3.2055 \(\int \frac{(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{2 (d+e x)^{3/2}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{4 \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-4*(c*d^2 - a*e^2)*Sqrt[d + e*x])/(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2]) + (2*(d + e*x)^(3/2))/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.184802, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{2 (d+e x)^{3/2}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{4 \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(c*d^2 - a*e^2)*Sqrt[d + e*x])/(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2]) + (2*(d + e*x)^(3/2))/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi in Sympy [A]  time = 33.4276, size = 97, normalized size = 0.92 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{4 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )}{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**(3/2)/(c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 4*sqrt
(d + e*x)*(a*e**2 - c*d**2)/(c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d
**2)))

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Mathematica [A]  time = 0.0518206, size = 51, normalized size = 0.49 \[ -\frac{2 \sqrt{d+e x} \left (c d (d-e x)-2 a e^2\right )}{c^2 d^2 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(-2*a*e^2 + c*d*(d - e*x)))/(c^2*d^2*Sqrt[(a*e + c*d*x)*(d + e
*x)])

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Maple [A]  time = 0.006, size = 68, normalized size = 0.7 \[ 2\,{\frac{ \left ( cdx+ae \right ) \left ( cdex+2\,a{e}^{2}-c{d}^{2} \right ) \left ( ex+d \right ) ^{3/2}}{{c}^{2}{d}^{2} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.82564, size = 49, normalized size = 0.47 \[ \frac{2 \,{\left (c d e x - c d^{2} + 2 \, a e^{2}\right )}}{\sqrt{c d x + a e} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d*e*x - c*d^2 + 2*a*e^2)/(sqrt(c*d*x + a*e)*c^2*d^2)

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Fricas [A]  time = 0.216694, size = 97, normalized size = 0.92 \[ \frac{2 \,{\left (c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2}\right )}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*sqrt(e*x + d)*c^2*d^2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.633167, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x