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

Optimal. Leaf size=160 \[ -\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}} \]

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(5/2))/(1155*c*e*(d + e*x)^(5/2)) - (64*d^2*(c*d^2
- c*e^2*x^2)^(5/2))/(231*c*e*(d + e*x)^(3/2)) - (8*d*(c*d^2 - c*e^2*x^2)^(5/2))
/(33*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(5/2))/(11*c*e)

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Rubi [A]  time = 0.229985, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(5/2))/(1155*c*e*(d + e*x)^(5/2)) - (64*d^2*(c*d^2
- c*e^2*x^2)^(5/2))/(231*c*e*(d + e*x)^(3/2)) - (8*d*(c*d^2 - c*e^2*x^2)^(5/2))
/(33*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(5/2))/(11*c*e)

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Rubi in Sympy [A]  time = 21.8676, size = 138, normalized size = 0.86 \[ - \frac{256 d^{3} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{1155 c e \left (d + e x\right )^{\frac{5}{2}}} - \frac{64 d^{2} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{231 c e \left (d + e x\right )^{\frac{3}{2}}} - \frac{8 d \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{33 c e \sqrt{d + e x}} - \frac{2 \sqrt{d + e x} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{11 c e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-256*d**3*(c*d**2 - c*e**2*x**2)**(5/2)/(1155*c*e*(d + e*x)**(5/2)) - 64*d**2*(c
*d**2 - c*e**2*x**2)**(5/2)/(231*c*e*(d + e*x)**(3/2)) - 8*d*(c*d**2 - c*e**2*x*
*2)**(5/2)/(33*c*e*sqrt(d + e*x)) - 2*sqrt(d + e*x)*(c*d**2 - c*e**2*x**2)**(5/2
)/(11*c*e)

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Mathematica [A]  time = 0.0611923, size = 73, normalized size = 0.46 \[ -\frac{2 c (d-e x)^2 \left (533 d^3+755 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{1155 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)*(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)]*(533*d^3 + 755*d^2*e*x + 455*d*e^2*x^2
+ 105*e^3*x^3))/(1155*e*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 66, normalized size = 0.4 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 105\,{e}^{3}{x}^{3}+455\,d{e}^{2}{x}^{2}+755\,{d}^{2}xe+533\,{d}^{3} \right ) }{1155\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(-e*x+d)*(105*e^3*x^3+455*d*e^2*x^2+755*d^2*e*x+533*d^3)*(-c*e^2*x^2+c*d
^2)^(3/2)/e/(e*x+d)^(3/2)

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Maxima [A]  time = 0.73902, size = 130, normalized size = 0.81 \[ -\frac{2 \,{\left (105 \, c^{\frac{3}{2}} e^{5} x^{5} + 245 \, c^{\frac{3}{2}} d e^{4} x^{4} - 50 \, c^{\frac{3}{2}} d^{2} e^{3} x^{3} - 522 \, c^{\frac{3}{2}} d^{3} e^{2} x^{2} - 311 \, c^{\frac{3}{2}} d^{4} e x + 533 \, c^{\frac{3}{2}} d^{5}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{1155 \,{\left (e^{2} x + d e\right )}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

-2/1155*(105*c^(3/2)*e^5*x^5 + 245*c^(3/2)*d*e^4*x^4 - 50*c^(3/2)*d^2*e^3*x^3 -
522*c^(3/2)*d^3*e^2*x^2 - 311*c^(3/2)*d^4*e*x + 533*c^(3/2)*d^5)*(e*x + d)*sqrt(
-e*x + d)/(e^2*x + d*e)

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Fricas [A]  time = 0.215543, size = 171, normalized size = 1.07 \[ \frac{2 \,{\left (105 \, c^{2} e^{7} x^{7} + 245 \, c^{2} d e^{6} x^{6} - 155 \, c^{2} d^{2} e^{5} x^{5} - 767 \, c^{2} d^{3} e^{4} x^{4} - 261 \, c^{2} d^{4} e^{3} x^{3} + 1055 \, c^{2} d^{5} e^{2} x^{2} + 311 \, c^{2} d^{6} e x - 533 \, c^{2} d^{7}\right )}}{1155 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/1155*(105*c^2*e^7*x^7 + 245*c^2*d*e^6*x^6 - 155*c^2*d^2*e^5*x^5 - 767*c^2*d^3*
e^4*x^4 - 261*c^2*d^4*e^3*x^3 + 1055*c^2*d^5*e^2*x^2 + 311*c^2*d^6*e*x - 533*c^2
*d^7)/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-c*(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-c*e^2*x^2 + c*d^2)^(3/2)*(e*x + d)^(3/2), x)