3.17.35 \(\int \frac {(b+2 c x) (a+b x+c x^2)^{3/2}}{(d+e x)^{5/2}} \, dx\) [1635]

Optimal. Leaf size=573 \[ -\frac {2 \left (128 c^2 d^2+15 b^2 e^2-4 c e (28 b d-9 a e)+16 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^5 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+15 b^2 e^2-4 c e (32 b d-17 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

2/15*(6*c*e*x-5*b*e+16*c*d)*(c*x^2+b*x+a)^(3/2)/e^2/(e*x+d)^(3/2)-2/15*(128*c^2*d^2+15*b^2*e^2-4*c*e*(-9*a*e+2
8*b*d)+16*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/e^4/(e*x+d)^(1/2)+2/15*(128*c^2*d^2+23*b^2*e^2-4*c*e*(-9*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)/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)-2/15*(-b*e+2*c*d)*(128
*c^2*d^2+15*b^2*e^2-4*c*e*(-17*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)

________________________________________________________________________________________

Rubi [A]
time = 0.39, antiderivative size = 573, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {826, 857, 732, 435, 430} \begin {gather*} -\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \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-17 a e)+15 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 )}{15 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 \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}} \left (-4 c e (32 b d-9 a e)+23 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 )}{15 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 {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{15 e^4 \sqrt {d+e x}}+\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^(5/2),x]

[Out]

(-2*(128*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(28*b*d - 9*a*e) + 16*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(15*e^
4*Sqrt[d + e*x]) + (2*(16*c*d - 5*b*e + 6*c*e*x)*(a + b*x + c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) + (2*Sqrt[2
]*Sqrt[b^2 - 4*a*c]*(128*c^2*d^2 + 23*b^2*e^2 - 4*c*e*(32*b*d - 9*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*Sq
rt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(15*e^5*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]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 15*b^2*e^2 - 4*c
*e*(32*b*d - 17*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)])/(15*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 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)^{5/2}} \, dx &=\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int \frac {\left (\frac {1}{2} \left (16 b c d-5 b^2 e-12 a c e\right )+8 c (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^2}\\ &=-\frac {2 \left (128 c^2 d^2+15 b^2 e^2-4 c e (28 b d-9 a e)+16 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {4 \int \frac {\frac {1}{4} \left (-112 b^2 c d e-64 a c^2 d e+15 b^3 e^2+4 b c \left (32 c d^2+17 a e^2\right )\right )+\frac {1}{2} c \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^4}\\ &=-\frac {2 \left (128 c^2 d^2+15 b^2 e^2-4 c e (28 b d-9 a e)+16 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2+15 b^2 e^2-4 c e (32 b d-17 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 e^5}+\frac {\left (2 c \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 e^5}\\ &=-\frac {2 \left (128 c^2 d^2+15 b^2 e^2-4 c e (28 b d-9 a e)+16 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 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 )}{15 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 (2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+15 b^2 e^2-4 c e (32 b d-17 a e)\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 )}{15 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \left (128 c^2 d^2+15 b^2 e^2-4 c e (28 b d-9 a e)+16 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 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 )}{15 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 {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+15 b^2 e^2-4 c e (32 b d-17 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 )}{15 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 14.45, size = 1186, normalized size = 2.07 \begin {gather*} \frac {\sqrt {d+e x} \left (-\frac {2 (a+x (b+c x)) \left (5 b e^2 (3 b d+a e+4 b e x)+2 c^2 \left (64 d^3+80 d^2 e x+8 d e^2 x^2-3 e^3 x^3\right )+c e \left (10 a e (2 d+3 e x)-b \left (112 d^2+144 d e x+17 e^2 x^2\right )\right )\right )}{e^4 (d+e x)^2}-\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 (36 a^2 c e^2+\left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) x (b+c x)+a \left (23 b^2 e^2+4 b c e (-32 d+9 e x)+4 c^2 \left (32 d^2+9 e^2 x^2\right )\right )\right )+i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (128 c^2 d^2+23 b^2 e^2+4 c e (-32 b d+9 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 (8 b^3 e^3-b^2 e^2 \left (16 c d+23 \sqrt {\left (b^2-4 a c\right ) e^2}\right )-32 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )-4 c \left (32 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-16 c d+9 \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 )}{e^6 \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}}} (d+e x)}\right )}{15 \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)^(5/2),x]

[Out]

(Sqrt[d + e*x]*((-2*(a + x*(b + c*x))*(5*b*e^2*(3*b*d + a*e + 4*b*e*x) + 2*c^2*(64*d^3 + 80*d^2*e*x + 8*d*e^2*
x^2 - 3*e^3*x^3) + c*e*(10*a*e*(2*d + 3*e*x) - b*(112*d^2 + 144*d*e*x + 17*e^2*x^2))))/(e^4*(d + e*x)^2) - (-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])]*(36*a^2*c*e^2 + (128*c^2*d^2 -
128*b*c*d*e + 23*b^2*e^2)*x*(b + c*x) + a*(23*b^2*e^2 + 4*b*c*e*(-32*d + 9*e*x) + 4*c^2*(32*d^2 + 9*e^2*x^2)))
+ I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(128*c^2*d^2 + 23*b^2*e^2 + 4*c*e*(-32*b*d + 9*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 + 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 + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Ellipti
cE[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]*(8*b^3*e^3 -
b^2*e^2*(16*c*d + 23*Sqrt[(b^2 - 4*a*c)*e^2]) - 32*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) - 4*c*(32*c*d
^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-16*c*d + 9*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 + Sqrt[(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]))])/(e^6*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sq
rt[(b^2 - 4*a*c)*e^2])]*(d + e*x))))/(15*Sqrt[a + x*(b + c*x)])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(9598\) vs. \(2(509)=1018\).
time = 1.17, size = 9599, normalized size = 16.75

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1402\)
risch	\(\text {Expression too large to display}\)	\(2840\)
default	\(\text {Expression too large to display}\)	\(9599\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

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)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)/(x*e + d)^(5/2), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.89, size = 796, normalized size = 1.39 \begin {gather*} -\frac {2 \, {\left ({\left (256 \, c^{3} d^{5} + {\left (b^{3} - 132 \, a b c\right )} x^{2} e^{5} + 2 \, {\left (3 \, {\left (21 \, b^{2} c + 44 \, a c^{2}\right )} d x^{2} + {\left (b^{3} - 132 \, a b c\right )} d x\right )} e^{4} - {\left (384 \, b c^{2} d^{2} x^{2} - 12 \, {\left (21 \, b^{2} c + 44 \, a c^{2}\right )} d^{2} x - {\left (b^{3} - 132 \, a b c\right )} d^{2}\right )} e^{3} + 2 \, {\left (128 \, c^{3} d^{3} x^{2} - 384 \, b c^{2} d^{3} x + 3 \, {\left (21 \, b^{2} c + 44 \, a c^{2}\right )} d^{3}\right )} e^{2} + 128 \, {\left (4 \, c^{3} d^{4} x - 3 \, b c^{2} 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 ) + 6 \, {\left (128 \, c^{3} d^{4} e + {\left (23 \, b^{2} c + 36 \, a c^{2}\right )} x^{2} e^{5} - 2 \, {\left (64 \, b c^{2} d x^{2} - {\left (23 \, b^{2} c + 36 \, a c^{2}\right )} d x\right )} e^{4} + {\left (128 \, c^{3} d^{2} x^{2} - 256 \, b c^{2} d^{2} x + {\left (23 \, b^{2} c + 36 \, a c^{2}\right )} d^{2}\right )} e^{3} + 128 \, {\left (2 \, c^{3} d^{3} x - b c^{2} 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^{3} d^{3} e^{2} - {\left (6 \, c^{3} x^{3} + 17 \, b c^{2} x^{2} - 5 \, a b c - 10 \, {\left (2 \, b^{2} c + 3 \, a c^{2}\right )} x\right )} e^{5} + {\left (16 \, c^{3} d x^{2} - 144 \, b c^{2} d x + 5 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d\right )} e^{4} + 16 \, {\left (10 \, c^{3} d^{2} x - 7 \, b c^{2} d^{2}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )}}{45 \, {\left (c x^{2} e^{8} + 2 \, c d x e^{7} + c d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

-2/45*((256*c^3*d^5 + (b^3 - 132*a*b*c)*x^2*e^5 + 2*(3*(21*b^2*c + 44*a*c^2)*d*x^2 + (b^3 - 132*a*b*c)*d*x)*e^
4 - (384*b*c^2*d^2*x^2 - 12*(21*b^2*c + 44*a*c^2)*d^2*x - (b^3 - 132*a*b*c)*d^2)*e^3 + 2*(128*c^3*d^3*x^2 - 38
4*b*c^2*d^3*x + 3*(21*b^2*c + 44*a*c^2)*d^3)*e^2 + 128*(4*c^3*d^4*x - 3*b*c^2*d^4)*e)*sqrt(c)*e^(1/2)*weierstr
assPInverse(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*(128*c^3*d^4*e
+ (23*b^2*c + 36*a*c^2)*x^2*e^5 - 2*(64*b*c^2*d*x^2 - (23*b^2*c + 36*a*c^2)*d*x)*e^4 + (128*c^3*d^2*x^2 - 256
*b*c^2*d^2*x + (23*b^2*c + 36*a*c^2)*d^2)*e^3 + 128*(2*c^3*d^3*x - b*c^2*d^3)*e^2)*sqrt(c)*e^(1/2)*weierstrass
Zeta(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^3*d^3*e^2 - (6*c^3*x^3 + 17*b*c^2*x^2 - 5*a*b*c - 10*(2*b^2
*c + 3*a*c^2)*x)*e^5 + (16*c^3*d*x^2 - 144*b*c^2*d*x + 5*(3*b^2*c + 4*a*c^2)*d)*e^4 + 16*(10*c^3*d^2*x - 7*b*c
^2*d^2)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))/(c*x^2*e^8 + 2*c*d*x*e^7 + c*d^2*e^6)

________________________________________________________________________________________

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 {5}{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)**(5/2),x)

[Out]

Integral((b + 2*c*x)*(a + b*x + c*x**2)**(3/2)/(d + e*x)**(5/2), x)

________________________________________________________________________________________

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)^(5/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)/(x*e + d)^(5/2), x)

________________________________________________________________________________________

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 )}^{5/2}} \,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)^(5/2),x)

[Out]

int(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^(5/2), x)

________________________________________________________________________________________