3.17.34 \(\int \frac {(b+2 c x) (a+b x+c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [1634]

Optimal. Leaf size=592 \[ \frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 e^4}+\frac {2 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

2/7*(2*c*e*x-7*b*e+16*c*d)*(c*x^2+b*x+a)^(3/2)/e^2/(e*x+d)^(1/2)+2/35*(128*c^2*d^2+51*b^2*e^2-4*c*e*(-5*a*e+44
*b*d)-48*c*e*(-b*e+2*c*d)*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/e^4-1/35*(-b*e+2*c*d)*(128*c^2*d^2+3*b^2*e^2-4*
c*e*(-29*a*e+32*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)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*
x+a)/(-4*a*c+b^2))^(1/2)/c/e^5/(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)+4/35*(a*
e^2-b*d*e+c*d^2)*(128*c^2*d^2+27*b^2*e^2-4*c*e*(-5*a*e+32*b*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)*(-4*
a*c+b^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)/c/e^5/
(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.46, antiderivative size = 592, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {826, 828, 857, 732, 435, 430} \begin {gather*} \frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \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 )}{35 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 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 )}{35 c e^5 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{35 e^4}+\frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 + 51*b^2*e^2 - 4*c*e*(44*b*d - 5*a*e) - 48*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c
*x^2])/(35*e^4) + (2*(16*c*d - 7*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) - (Sqrt[2]*Sqrt
[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(32*b*d - 29*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]]/Sqrt[2]]
, (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(35*c*e^5*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^
2 + 27*b^2*e^2 - 4*c*e*(32*b*d - 5*a*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))]*EllipticF[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)])/(35*c*e^5*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 826

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

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (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 {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=\frac {2 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {6 \int \frac {\left (\frac {1}{2} \left (16 b c d-7 b^2 e-4 a c e\right )+8 c (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^2}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 e^4}+\frac {2 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {4 \int \frac {-\frac {1}{4} c \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{35 c e^4}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 e^4}+\frac {2 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{35 e^5}+\frac {\left (4 \left (-\frac {1}{4} c e \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )+\frac {1}{4} c d (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{35 c e^5}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 e^4}+\frac {2 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 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 )}{35 c e^5 \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 (8 \sqrt {2} \sqrt {b^2-4 a c} \left (-\frac {1}{4} c e \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )+\frac {1}{4} c d (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right ) \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 )}{35 c^2 e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 e^4}+\frac {2 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 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 )}{35 c e^5 \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 {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2+20 a c e^2\right ) \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 )}{35 c e^5 \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 = 14.70, size = 884, normalized size = 1.49 \begin {gather*} \frac {\frac {4 e^2 (-2 c d+b e) \left (128 c^2 d^2+3 b^2 e^2+4 c e (-32 b d+29 a e)\right ) (a+x (b+c x))}{c \sqrt {d+e x}}+\frac {4 e^2 (a+x (b+c x)) \left (b e^2 (51 b d-35 a e+16 b e x)+2 c^2 \left (64 d^3+16 d^2 e x-8 d e^2 x^2+5 e^3 x^3\right )+c e \left (10 a e (10 d+3 e x)+b \left (-176 d^2-48 d e x+23 e^2 x^2\right )\right )\right )}{\sqrt {d+e x}}+\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (-\left ((-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (128 c^2 d^2+3 b^2 e^2+4 c e (-32 b d+29 a e)\right ) 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 )\right )+\left (-3 b^4 e^4+b^3 e^3 \left (32 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )-2 b^2 c e^2 \left (16 c d^2+4 a e^2+67 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+8 c^2 \left (10 a^2 e^4-32 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (16 c d-29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+4 b c e \left (96 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-32 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) 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 )\right )}{c \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}}}}}{70 e^6 \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*e^2*(-2*c*d + b*e)*(128*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-32*b*d + 29*a*e))*(a + x*(b + c*x)))/(c*Sqrt[d + e*x
]) + (4*e^2*(a + x*(b + c*x))*(b*e^2*(51*b*d - 35*a*e + 16*b*e*x) + 2*c^2*(64*d^3 + 16*d^2*e*x - 8*d*e^2*x^2 +
5*e^3*x^3) + c*e*(10*a*e*(10*d + 3*e*x) + b*(-176*d^2 - 48*d*e*x + 23*e^2*x^2))))/Sqrt[d + e*x] + (I*(d + e*x
)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*
d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*(-((-2*c*d + b*e)*(2*c*d - b*e
+ Sqrt[(b^2 - 4*a*c)*e^2])*(128*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-32*b*d + 29*a*e))*EllipticE[I*ArcSinh[(Sqrt[2]*S
qrt[(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]))]) + (-3*b^4*e^4 + b^3*e^3*(32*c*d + 3*Sqrt[(b^2 -
4*a*c)*e^2]) - 2*b^2*c*e^2*(16*c*d^2 + 4*a*e^2 + 67*d*Sqrt[(b^2 - 4*a*c)*e^2]) + 8*c^2*(10*a^2*e^4 - 32*c*d^3
*Sqrt[(b^2 - 4*a*c)*e^2] + a*d*e^2*(16*c*d - 29*Sqrt[(b^2 - 4*a*c)*e^2])) + 4*b*c*e*(96*c*d^2*Sqrt[(b^2 - 4*a*
c)*e^2] + a*e^2*(-32*c*d + 29*Sqrt[(b^2 - 4*a*c)*e^2])))*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]))]))/(c*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)
*e^2])]))/(70*e^6*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. \(6526\) vs. \(2(528)=1056\).
time = 1.25, size = 6527, normalized size = 11.03

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1720\)
risch	\(\text {Expression too large to display}\)	\(1897\)
default	\(\text {Expression too large to display}\)	\(6527\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)/(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 = 0.54, size = 801, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left ({\left (256 \, c^{4} d^{5} - {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} x e^{5} - {\left (2 \, {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d x + {\left (3 \, b^{4} - 46 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d\right )} e^{4} + 2 \, {\left ({\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{2} x - {\left (11 \, b^{3} c + 212 \, a b c^{2}\right )} d^{2}\right )} e^{3} - 2 \, {\left (256 \, b c^{3} d^{3} x - {\left (139 \, b^{2} c^{2} + 212 \, a c^{3}\right )} d^{3}\right )} e^{2} + 256 \, {\left (c^{4} d^{4} x - 2 \, b c^{3} d^{4}\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 ) + 3 \, {\left (256 \, c^{4} d^{4} e - {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} x e^{5} + {\left (2 \, {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d x - {\left (3 \, b^{3} c + 116 \, a b c^{2}\right )} d\right )} e^{4} - 2 \, {\left (192 \, b c^{3} d^{2} x - {\left (67 \, b^{2} c^{2} + 116 \, a c^{3}\right )} d^{2}\right )} e^{3} + 128 \, {\left (2 \, c^{4} d^{3} x - 3 \, b c^{3} d^{3}\right )} e^{2}\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 \, {\left (128 \, c^{4} d^{3} e^{2} + {\left (10 \, c^{4} x^{3} + 23 \, b c^{3} x^{2} - 35 \, a b c^{2} + 2 \, {\left (8 \, b^{2} c^{2} + 15 \, a c^{3}\right )} x\right )} e^{5} - {\left (16 \, c^{4} d x^{2} + 48 \, b c^{3} d x - {\left (51 \, b^{2} c^{2} + 100 \, a c^{3}\right )} d\right )} e^{4} + 16 \, {\left (2 \, c^{4} d^{2} x - 11 \, b c^{3} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{105 \, {\left (c^{2} x e^{7} + c^{2} d e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*((256*c^4*d^5 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*x*e^5 - (2*(11*b^3*c + 212*a*b*c^2)*d*x + (3*b^4 - 46
*a*b^2*c - 120*a^2*c^2)*d)*e^4 + 2*((139*b^2*c^2 + 212*a*c^3)*d^2*x - (11*b^3*c + 212*a*b*c^2)*d^2)*e^3 - 2*(2
56*b*c^3*d^3*x - (139*b^2*c^2 + 212*a*c^3)*d^3)*e^2 + 256*(c^4*d^4*x - 2*b*c^3*d^4)*e)*sqrt(c)*e^(1/2)*weierst
rassPInverse(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*(256*c^4*d^4*
e - (3*b^3*c + 116*a*b*c^2)*x*e^5 + (2*(67*b^2*c^2 + 116*a*c^3)*d*x - (3*b^3*c + 116*a*b*c^2)*d)*e^4 - 2*(192*
b*c^3*d^2*x - (67*b^2*c^2 + 116*a*c^3)*d^2)*e^3 + 128*(2*c^4*d^3*x - 3*b*c^3*d^3)*e^2)*sqrt(c)*e^(1/2)*weierst
rassZeta(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*(128*c^4*d^3*e^2 + (10*c^4*x^3 + 23*b*c^3*x^2 - 35*a*b*c^2 + 2
*(8*b^2*c^2 + 15*a*c^3)*x)*e^5 - (16*c^4*d*x^2 + 48*b*c^3*d*x - (51*b^2*c^2 + 100*a*c^3)*d)*e^4 + 16*(2*c^4*d^
2*x - 11*b*c^3*d^2)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))/(c^2*x*e^7 + c^2*d*e^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)/(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 {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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