3.858 \(\int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=201 \[ -\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}-\frac{4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac{1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}} \]

[Out]

(-4096*d^4*(c*d^2 - c*e^2*x^2)^(5/2))/(15015*c*e*(d + e*x)^(5/2)) - (1024*d^3*(c
*d^2 - c*e^2*x^2)^(5/2))/(3003*c*e*(d + e*x)^(3/2)) - (128*d^2*(c*d^2 - c*e^2*x^
2)^(5/2))/(429*c*e*Sqrt[d + e*x]) - (32*d*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(5/2
))/(143*c*e) - (2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(5/2))/(13*c*e)

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Rubi [A]  time = 0.303693, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}-\frac{4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac{1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4096*d^4*(c*d^2 - c*e^2*x^2)^(5/2))/(15015*c*e*(d + e*x)^(5/2)) - (1024*d^3*(c
*d^2 - c*e^2*x^2)^(5/2))/(3003*c*e*(d + e*x)^(3/2)) - (128*d^2*(c*d^2 - c*e^2*x^
2)^(5/2))/(429*c*e*Sqrt[d + e*x]) - (32*d*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(5/2
))/(143*c*e) - (2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(5/2))/(13*c*e)

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Rubi in Sympy [A]  time = 29.0335, size = 173, normalized size = 0.86 \[ - \frac{4096 d^{4} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{15015 c e \left (d + e x\right )^{\frac{5}{2}}} - \frac{1024 d^{3} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{3003 c e \left (d + e x\right )^{\frac{3}{2}}} - \frac{128 d^{2} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{429 c e \sqrt{d + e x}} - \frac{32 d \sqrt{d + e x} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{143 c e} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{13 c e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4096*d**4*(c*d**2 - c*e**2*x**2)**(5/2)/(15015*c*e*(d + e*x)**(5/2)) - 1024*d**
3*(c*d**2 - c*e**2*x**2)**(5/2)/(3003*c*e*(d + e*x)**(3/2)) - 128*d**2*(c*d**2 -
c*e**2*x**2)**(5/2)/(429*c*e*sqrt(d + e*x)) - 32*d*sqrt(d + e*x)*(c*d**2 - c*e*
*2*x**2)**(5/2)/(143*c*e) - 2*(d + e*x)**(3/2)*(c*d**2 - c*e**2*x**2)**(5/2)/(13
*c*e)

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Mathematica [A]  time = 0.070875, size = 84, normalized size = 0.42 \[ -\frac{2 c (d-e x)^2 \left (9683 d^4+16700 d^3 e x+14210 d^2 e^2 x^2+6300 d e^3 x^3+1155 e^4 x^4\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{15015 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(9683*d^4 + 16700*d^3*e*x + 14210*d^2*
e^2*x^2 + 6300*d*e^3*x^3 + 1155*e^4*x^4))/(15015*e*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 77, normalized size = 0.4 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 1155\,{e}^{4}{x}^{4}+6300\,d{e}^{3}{x}^{3}+14210\,{d}^{2}{e}^{2}{x}^{2}+16700\,{d}^{3}xe+9683\,{d}^{4} \right ) }{15015\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15015*(-e*x+d)*(1155*e^4*x^4+6300*d*e^3*x^3+14210*d^2*e^2*x^2+16700*d^3*e*x+9
683*d^4)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

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Maxima [A]  time = 0.733559, size = 149, normalized size = 0.74 \[ -\frac{2 \,{\left (1155 \, c^{\frac{3}{2}} e^{6} x^{6} + 3990 \, c^{\frac{3}{2}} d e^{5} x^{5} + 2765 \, c^{\frac{3}{2}} d^{2} e^{4} x^{4} - 5420 \, c^{\frac{3}{2}} d^{3} e^{3} x^{3} - 9507 \, c^{\frac{3}{2}} d^{4} e^{2} x^{2} - 2666 \, c^{\frac{3}{2}} d^{5} e x + 9683 \, c^{\frac{3}{2}} d^{6}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{15015 \,{\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15015*(1155*c^(3/2)*e^6*x^6 + 3990*c^(3/2)*d*e^5*x^5 + 2765*c^(3/2)*d^2*e^4*x
^4 - 5420*c^(3/2)*d^3*e^3*x^3 - 9507*c^(3/2)*d^4*e^2*x^2 - 2666*c^(3/2)*d^5*e*x
+ 9683*c^(3/2)*d^6)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

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Fricas [A]  time = 0.213389, size = 190, normalized size = 0.95 \[ \frac{2 \,{\left (1155 \, c^{2} e^{8} x^{8} + 3990 \, c^{2} d e^{7} x^{7} + 1610 \, c^{2} d^{2} e^{6} x^{6} - 9410 \, c^{2} d^{3} e^{5} x^{5} - 12272 \, c^{2} d^{4} e^{4} x^{4} + 2754 \, c^{2} d^{5} e^{3} x^{3} + 19190 \, c^{2} d^{6} e^{2} x^{2} + 2666 \, c^{2} d^{7} e x - 9683 \, c^{2} d^{8}\right )}}{15015 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*c^2*e^8*x^8 + 3990*c^2*d*e^7*x^7 + 1610*c^2*d^2*e^6*x^6 - 9410*c^2
*d^3*e^5*x^5 - 12272*c^2*d^4*e^4*x^4 + 2754*c^2*d^5*e^3*x^3 + 19190*c^2*d^6*e^2*
x^2 + 2666*c^2*d^7*e*x - 9683*c^2*d^8)/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.431761, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done