3.2436 \(\int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=816 \[ \frac{2 e \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{1155 c^3 e^3}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 + 8*b^4*e^4 - c^3*d^2*e*(19*b*d - 42*a*e) - b^2*c*e^
3*(19*b*d + 21*a*e) + 3*c^2*e^2*(2*b^2*d^2 + 17*a*b*d*e - 10*a^2*e^2) - 3*c*e*(2
*c*d - b*e)*(c^2*d^2 + 8*b^2*e^2 - c*e*(b*d + 31*a*e))*x)*Sqrt[a + b*x + c*x^2])
/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 - 6*b^2*e^2 + c*e*(13*b*d - 3*a*e) +
14*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*
x]*(a + b*x + c*x^2)^(5/2))/(11*c) - (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*
(c^2*d^2 - 2*b^2*e^2 - c*e*(b*d - 9*a*e))*(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[Sq
rt[(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)])/(1155*c^4*e^4*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]*(c*d^2 - b*d*e + a*e^2)*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d
- 21*a*e) + b^2*c*e^3*(13*b*d + 51*a*e) + 3*c^2*e^2*(b^2*d^2 - 28*a*b*d*e - 20*a
^2*e^2))*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)])/(1155*c^4*e^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 5.58551, antiderivative size = 816, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 e \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{1155 c^3 e^3}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 + 8*b^4*e^4 - c^3*d^2*e*(19*b*d - 42*a*e) - b^2*c*e^
3*(19*b*d + 21*a*e) + 3*c^2*e^2*(2*b^2*d^2 + 17*a*b*d*e - 10*a^2*e^2) - 3*c*e*(2
*c*d - b*e)*(c^2*d^2 + 8*b^2*e^2 - c*e*(b*d + 31*a*e))*x)*Sqrt[a + b*x + c*x^2])
/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 - 6*b^2*e^2 + c*e*(13*b*d - 3*a*e) +
14*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*
x]*(a + b*x + c*x^2)^(5/2))/(11*c) - (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*
(c^2*d^2 - 2*b^2*e^2 - c*e*(b*d - 9*a*e))*(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[Sq
rt[(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)])/(1155*c^4*e^4*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]*(c*d^2 - b*d*e + a*e^2)*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d
- 21*a*e) + b^2*c*e^3*(13*b*d + 51*a*e) + 3*c^2*e^2*(b^2*d^2 - 28*a*b*d*e - 20*a
^2*e^2))*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)])/(1155*c^4*e^4*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((e*x+d)**(3/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 15.3526, size = 10848, normalized size = 13.29 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.088, size = 11933, normalized size = 14.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)*(e*x + d)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*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. \[ \int \left (d + e x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out