3.2470 \(\int \frac{1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=918 \[ -\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 e \sqrt{c x^2+b x+a} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt{d+e x}}+\frac{8 \sqrt{2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}-\frac{2 \left (5 a c e (2 c d-b e)^2-4 c \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}-\frac{2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \left (c x^2+b x+a\right )^{3/2}} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d
*e + a*e^2)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2)) - (2*(5*a*c*e*(2*c*d - b*e)^2
- (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 4*b^2*e^2 - c*e*(3*b*d - 14*a*e)) - 4*
c*(2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*x))/(3*(b^2 - 4*a*c)
^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]) + (2*e*(16*c^4
*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3
*c^2*e^2*(3*b^2*d^2 - 20*a*b*d*e - 28*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) - (Sqrt[2]*(16*c^4*d^4 - 8*b^
4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3*c^2*e^2*(3
*b^2*d^2 - 20*a*b*d*e - 28*a^2*e^2))*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)]
)/(3*(b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (8*Sqrt[2]*(2*c*d - b*e)*(2*
c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*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[S
qrt[(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)])/(3*(b^2 - 4*a*c)^(3/2)*(c*d^2 -
b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 4.13448, antiderivative size = 918, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 e \sqrt{c x^2+b x+a} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt{d+e x}}+\frac{8 \sqrt{2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}-\frac{2 \left (5 a c e (2 c d-b e)^2-4 c \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}-\frac{2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \left (c x^2+b x+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d
*e + a*e^2)*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2)) - (2*(5*a*c*e*(2*c*d - b*e)^2
- (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 4*b^2*e^2 - c*e*(3*b*d - 14*a*e)) - 4*
c*(2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*x))/(3*(b^2 - 4*a*c)
^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]) + (2*e*(16*c^4
*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3
*c^2*e^2*(3*b^2*d^2 - 20*a*b*d*e - 28*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) - (Sqrt[2]*(16*c^4*d^4 - 8*b^
4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3*c^2*e^2*(3
*b^2*d^2 - 20*a*b*d*e - 28*a^2*e^2))*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)]
)/(3*(b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (
b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (8*Sqrt[2]*(2*c*d - b*e)*(2*
c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*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[S
qrt[(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)])/(3*(b^2 - 4*a*c)^(3/2)*(c*d^2 -
b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 15.3664, size = 7870, normalized size = 8.57 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

Result too large to show

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Maple [B]  time = 0.272, size = 27157, normalized size = 29.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(5/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c^{2} e x^{5} +{\left (c^{2} d + 2 \, b c e\right )} x^{4} +{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{3} + a^{2} d +{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} +{\left (2 \, a b d + a^{2} e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((c^2*e*x^5 + (c^2*d + 2*b*c*e)*x^4 + (2*b*c*d + (b^2 + 2*a*c)*e)*x^3
+ a^2*d + (2*a*b*e + (b^2 + 2*a*c)*d)*x^2 + (2*a*b*d + a^2*e)*x)*sqrt(c*x^2 + b
*x + a)*sqrt(e*x + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)**(3/2)*(a + b*x + c*x**2)**(5/2)), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out