3.2062 \(\int \frac{(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=107 \[ \frac{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac{2 (d+e x)^{5/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(4*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2)) - (2*(d + e*x)^(5/2))/(c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)
^(3/2))

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Rubi [A]  time = 0.198601, antiderivative size = 107, 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 \[ \frac{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac{2 (d+e x)^{5/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(4*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2)) - (2*(d + e*x)^(5/2))/(c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)
^(3/2))

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Rubi in Sympy [A]  time = 33.7382, size = 100, normalized size = 0.93 \[ - \frac{2 \left (d + e x\right )^{\frac{5}{2}}}{c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )}{3 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

-2*(d + e*x)**(5/2)/(c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)) - 4*
(d + e*x)**(3/2)*(a*e**2 - c*d**2)/(3*c**2*d**2*(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2))**(3/2))

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Mathematica [A]  time = 0.076003, size = 53, normalized size = 0.5 \[ -\frac{2 (d+e x)^{3/2} \left (2 a e^2+c d (d+3 e x)\right )}{3 c^2 d^2 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 68, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3\,cdex+2\,a{e}^{2}+c{d}^{2} \right ) }{3\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-2/3*(c*d*x+a*e)*(3*c*d*e*x+2*a*e^2+c*d^2)*(e*x+d)^(5/2)/c^2/d^2/(c*d*e*x^2+a*e^
2*x+c*d^2*x+a*d*e)^(5/2)

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Maxima [A]  time = 0.818572, size = 68, normalized size = 0.64 \[ -\frac{2 \,{\left (3 \, c d e x + c d^{2} + 2 \, a e^{2}\right )}}{3 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt{c d x + a e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21286, size = 177, normalized size = 1.65 \[ -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d e x + c d^{2} + 2 \, a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (c^{4} d^{4} e x^{3} + a^{2} c^{2} d^{3} e^{2} +{\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} x^{2} +{\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*e*x + c*d^2 + 2*a*e^2)*s
qrt(e*x + d)/(c^4*d^4*e*x^3 + a^2*c^2*d^3*e^2 + (c^4*d^5 + 2*a*c^3*d^3*e^2)*x^2
+ (2*a*c^3*d^4*e + a^2*c^2*d^2*e^3)*x)

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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)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.709031, 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)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x