3.25.74 \(\int \frac {1}{(d+e x)^{3/2} (a+b x+c x^2)^{3/2}} \, dx\) [2474]

Optimal. Leaf size=607 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {2 \sqrt {2} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-4
*e*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)+2*(c
^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),
(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*
a*c+b^2))^(1/2)/(a*e^2-b*d*e+c*d^2)^2/(-4*a*c+b^2)^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^
2)^(1/2))))^(1/2)-2*(-b*e+2*c*d)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)
,(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/
2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*
x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 607, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {754, 848, 857, 732, 435, 430} \begin {gather*} \frac {2 \sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {4 e \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a
+ b*x + c*x^2]) - (4*e*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 -
b*d*e + a*e^2)^2*Sqrt[d + e*x]) + (2*Sqrt[2]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*
(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sq
rt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e
^2)^2*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*(2*c*d - b*e
)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipti
cF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} e \left (b c d-2 b^2 e+6 a c e\right )+\frac {1}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} c e \left (b c d^2+b^2 d e-8 a c d e+a b e^2\right )+\frac {1}{2} c e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {(c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}+\frac {\left (2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (2 \sqrt {2} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {2 \sqrt {2} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 27.23, size = 1211, normalized size = 2.00 \begin {gather*} \frac {4 e^2 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (-3 a^2 c e^2+\left (c^2 d^2-b c d e+b^2 e^2\right ) x (b+c x)+a \left (b^2 e^2-b c e (d+3 e x)+c^2 \left (d^2-3 e^2 x^2\right )\right )\right )-2 e \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (\left (b^2-4 a c\right ) e^3 (a+x (b+c x))+(d+e x) \left (b^3 e^2+b^2 c e (-2 d+e x)+b c \left (-3 a e^2+c d (d-2 e x)\right )+2 c^2 \left (c d^2 x+a e (2 d-e x)\right )\right )\right )-i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-i \sqrt {2} \left (b^3 e^3-b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b c e \left (-4 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c \left (-c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (8 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\left (b^2-4 a c\right ) e \left (c d^2+e (-b d+a e)\right )^2 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x} \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(4*e^2*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(-3*a^2*c*e^2 + (c^2*d^2 - b*
c*d*e + b^2*e^2)*x*(b + c*x) + a*(b^2*e^2 - b*c*e*(d + 3*e*x) + c^2*(d^2 - 3*e^2*x^2))) - 2*e*Sqrt[(c*d^2 + e*
(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*((b^2 - 4*a*c)*e^3*(a + x*(b + c*x)) + (d + e*x)*(b^
3*e^2 + b^2*c*e*(-2*d + e*x) + b*c*(-3*a*e^2 + c*d*(d - 2*e*x)) + 2*c^2*(c*d^2*x + a*e*(2*d - e*x)))) - I*Sqrt
[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(d + e*x)^(3/2)*Sqrt[(-2*a
*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sq
rt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c
)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(Sqrt[2]*
Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt
[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - I*Sqrt[2]*(b^3*e^3 - b^2*e^2*(2*c*d + Sqrt[(b
^2 - 4*a*c)*e^2]) + b*c*e*(-4*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e^2
*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*(d + e*x)^(3/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x
+ e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*
e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + S
qrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e +
Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2
- 4*a*c)*e^2]))])/((b^2 - 4*a*c)*e*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e
+ Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4414\) vs. \(2(551)=1102\).
time = 0.98, size = 4415, normalized size = 7.27

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1849\)
default	\(\text {Expression too large to display}\)	\(4415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

(2*d^3*e*b*c^2+3*(-4*a*c+b^2)^(1/2)*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4
*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/
2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(
1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b*c*d^2*e^2-6*a*b*c*d*e^3+2*b^3*d*e^3+4*b^3*e^4*x-4*2
^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a
*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2
)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+
b^2)^(1/2)))^(1/2))*b^3*d*e^3+12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*
c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+
b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2
)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*c*e^4+3*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*
e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2
)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c
*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*b^3*d*e^3-12*2^(1/2)*(-
(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(
1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+
d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2
)))^(1/2))*a^2*c*e^4+4*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/
2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)
)^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*
d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b^2*e^4-3*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^
(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e
/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)
,(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b^2*e^4-8*e^4*a^2*c-2*b*c^2*d^2
*e^2*x-2*(-4*a*c+b^2)^(1/2)*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2
)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2
*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e
-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*c*d*e^3+4*c^3*d^2*e^2*x^2+4*a*c^2*d*e^3*x-12*2^(1/2)*(-(e*x
+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)
))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c
/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^
(1/2))*a*b*c*d*e^3+8*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2)
)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^
(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)
/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*d*e^3+2*a*b^2*e^4-12*a*c^2*e^4*x^2+4*b^2*c*e^4*x^2+4*2^(1/2)*(
-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^
(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x
+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/
2)))^(1/2))*c^3*d^4-4*b*c^2*d*e^3*x^2+8*d^2*e^2*c^2*a+12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))
^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*
e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2
),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2))*a*c^2*d^2*e^2+(-4*a*c+b^2)^(1/2)
*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))*e/(2*c*d-b*e+e*(-4
*a*c+b^2)^(1/2)))^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1
/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/(2*c*d-b*e+e*(-4*a*
c+b^2)^(1/2)))^(1/2))*a*b*e^4-3*2^(1/2)*(-(e*x+...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/2)*(x*e + d)^(3/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.08, size = 1581, normalized size = 2.60 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, c^{4} d^{4} x^{2} + 2 \, b c^{3} d^{4} x + 2 \, a c^{3} d^{4} + {\left ({\left (2 \, b^{3} c - 9 \, a b c^{2}\right )} x^{3} + {\left (2 \, b^{4} - 9 \, a b^{2} c\right )} x^{2} + {\left (2 \, a b^{3} - 9 \, a^{2} b c\right )} x\right )} e^{4} - {\left (3 \, {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d x^{3} + {\left (b^{3} c - 9 \, a b c^{2}\right )} d x^{2} - 2 \, {\left (b^{4} - 6 \, a b^{2} c + 9 \, a^{2} c^{2}\right )} d x - {\left (2 \, a b^{3} - 9 \, a^{2} b c\right )} d\right )} e^{3} - 3 \, {\left (b c^{3} d^{2} x^{3} + 2 \, {\left (b^{2} c^{2} - 3 \, a c^{3}\right )} d^{2} x^{2} + {\left (b^{3} c - 5 \, a b c^{2}\right )} d^{2} x + {\left (a b^{2} c - 6 \, a^{2} c^{2}\right )} d^{2}\right )} e^{2} + {\left (2 \, c^{4} d^{3} x^{3} - b c^{3} d^{3} x^{2} - 3 \, a b c^{2} d^{3} - {\left (3 \, b^{2} c^{2} - 2 \, a c^{3}\right )} d^{3} x\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 6 \, {\left ({\left ({\left (b^{2} c^{2} - 3 \, a c^{3}\right )} x^{3} + {\left (b^{3} c - 3 \, a b c^{2}\right )} x^{2} + {\left (a b^{2} c - 3 \, a^{2} c^{2}\right )} x\right )} e^{4} - {\left (b c^{3} d x^{3} + 3 \, a c^{3} d x^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} d x - {\left (a b^{2} c - 3 \, a^{2} c^{2}\right )} d\right )} e^{3} + {\left (c^{4} d^{2} x^{3} - a b c^{2} d^{2} - {\left (b^{2} c^{2} - a c^{3}\right )} d^{2} x\right )} e^{2} + {\left (c^{4} d^{3} x^{2} + b c^{3} d^{3} x + a c^{3} d^{3}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, \sqrt {c x^{2} + b x + a} {\left ({\left (a b^{2} c - 4 \, a^{2} c^{2} + 2 \, {\left (b^{2} c^{2} - 3 \, a c^{3}\right )} x^{2} + {\left (2 \, b^{3} c - 7 \, a b c^{2}\right )} x\right )} e^{4} - {\left (2 \, b c^{3} d x^{2} + {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d x - {\left (b^{3} c - 3 \, a b c^{2}\right )} d\right )} e^{3} + {\left (2 \, c^{4} d^{2} x^{2} - b c^{3} d^{2} x - 2 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2}\right )} e^{2} + {\left (2 \, c^{4} d^{3} x + b c^{3} d^{3}\right )} e\right )} \sqrt {x e + d}\right )}}{3 \, {\left ({\left ({\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} x^{3} + {\left (a^{2} b^{3} c - 4 \, a^{3} b c^{2}\right )} x^{2} + {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x\right )} e^{6} - {\left (2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d x^{3} + {\left (2 \, a b^{4} c - 9 \, a^{2} b^{2} c^{2} + 4 \, a^{3} c^{3}\right )} d x^{2} + {\left (a^{2} b^{3} c - 4 \, a^{3} b c^{2}\right )} d x - {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} d\right )} e^{5} + {\left ({\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{2} x^{3} + {\left (b^{5} c - 4 \, a b^{3} c^{2}\right )} d^{2} x^{2} - {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{2} x - 2 \, {\left (a^{2} b^{3} c - 4 \, a^{3} b c^{2}\right )} d^{2}\right )} e^{4} - {\left (2 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x^{3} + {\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{2} - {\left (b^{5} c - 4 \, a b^{3} c^{2}\right )} d^{3} x - {\left (a b^{4} c - 2 \, a^{2} b^{2} c^{2} - 8 \, a^{3} c^{3}\right )} d^{3}\right )} e^{3} + {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{4} x^{3} - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{4} x^{2} - {\left (2 \, b^{4} c^{2} - 9 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d^{4} x - 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{4}\right )} e^{2} + {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{5} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{5} x + {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} d^{5}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/3*((2*c^4*d^4*x^2 + 2*b*c^3*d^4*x + 2*a*c^3*d^4 + ((2*b^3*c - 9*a*b*c^2)*x^3 + (2*b^4 - 9*a*b^2*c)*x^2 + (2
*a*b^3 - 9*a^2*b*c)*x)*e^4 - (3*(b^2*c^2 - 6*a*c^3)*d*x^3 + (b^3*c - 9*a*b*c^2)*d*x^2 - 2*(b^4 - 6*a*b^2*c + 9
*a^2*c^2)*d*x - (2*a*b^3 - 9*a^2*b*c)*d)*e^3 - 3*(b*c^3*d^2*x^3 + 2*(b^2*c^2 - 3*a*c^3)*d^2*x^2 + (b^3*c - 5*a
*b*c^2)*d^2*x + (a*b^2*c - 6*a^2*c^2)*d^2)*e^2 + (2*c^4*d^3*x^3 - b*c^3*d^3*x^2 - 3*a*b*c^2*d^3 - (3*b^2*c^2 -
2*a*c^3)*d^3*x)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2
, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d +
(3*c*x + b)*e)*e^(-1)/c) + 6*(((b^2*c^2 - 3*a*c^3)*x^3 + (b^3*c - 3*a*b*c^2)*x^2 + (a*b^2*c - 3*a^2*c^2)*x)*e
^4 - (b*c^3*d*x^3 + 3*a*c^3*d*x^2 - (b^3*c - 4*a*b*c^2)*d*x - (a*b^2*c - 3*a^2*c^2)*d)*e^3 + (c^4*d^2*x^3 - a*
b*c^2*d^2 - (b^2*c^2 - a*c^3)*d^2*x)*e^2 + (c^4*d^3*x^2 + b*c^3*d^3*x + a*c^3*d^3)*e)*sqrt(c)*e^(1/2)*weierstr
assZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c -
6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)
*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)
/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)) + 3*sqrt(c*x^2 + b*x + a)*((a*b^2*c - 4*a^2*c^2 + 2*(b^2*c^2 - 3*a*
c^3)*x^2 + (2*b^3*c - 7*a*b*c^2)*x)*e^4 - (2*b*c^3*d*x^2 + (b^2*c^2 - 2*a*c^3)*d*x - (b^3*c - 3*a*b*c^2)*d)*e^
3 + (2*c^4*d^2*x^2 - b*c^3*d^2*x - 2*(b^2*c^2 - 2*a*c^3)*d^2)*e^2 + (2*c^4*d^3*x + b*c^3*d^3)*e)*sqrt(x*e + d)
)/(((a^2*b^2*c^2 - 4*a^3*c^3)*x^3 + (a^2*b^3*c - 4*a^3*b*c^2)*x^2 + (a^3*b^2*c - 4*a^4*c^2)*x)*e^6 - (2*(a*b^3
*c^2 - 4*a^2*b*c^3)*d*x^3 + (2*a*b^4*c - 9*a^2*b^2*c^2 + 4*a^3*c^3)*d*x^2 + (a^2*b^3*c - 4*a^3*b*c^2)*d*x - (a
^3*b^2*c - 4*a^4*c^2)*d)*e^5 + ((b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^2*x^3 + (b^5*c - 4*a*b^3*c^2)*d^2*x^2 -
(a*b^4*c - 6*a^2*b^2*c^2 + 8*a^3*c^3)*d^2*x - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d^2)*e^4 - (2*(b^3*c^3 - 4*a*b*c^4)*
d^3*x^3 + (b^4*c^2 - 6*a*b^2*c^3 + 8*a^2*c^4)*d^3*x^2 - (b^5*c - 4*a*b^3*c^2)*d^3*x - (a*b^4*c - 2*a^2*b^2*c^2
- 8*a^3*c^3)*d^3)*e^3 + ((b^2*c^4 - 4*a*c^5)*d^4*x^3 - (b^3*c^3 - 4*a*b*c^4)*d^4*x^2 - (2*b^4*c^2 - 9*a*b^2*c
^3 + 4*a^2*c^4)*d^4*x - 2*(a*b^3*c^2 - 4*a^2*b*c^3)*d^4)*e^2 + ((b^2*c^4 - 4*a*c^5)*d^5*x^2 + (b^3*c^3 - 4*a*b
*c^4)*d^5*x + (a*b^2*c^3 - 4*a^2*c^4)*d^5)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/((d + e*x)**(3/2)*(a + b*x + c*x**2)**(3/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/2)*(x*e + d)^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x)

[Out]

int(1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)), x)

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