3.2038 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c^2 d^2 (d+e x)^{7/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}} \]

[Out]

(4*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(63*c^2*d^2*(d
+ e*x)^(7/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d*(d + e
*x)^(5/2))

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Rubi [A]  time = 0.191076, antiderivative size = 109, 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 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c^2 d^2 (d+e x)^{7/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(63*c^2*d^2*(d
+ e*x)^(7/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d*(d + e
*x)^(5/2))

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Rubi in Sympy [A]  time = 34.2268, size = 100, normalized size = 0.92 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{9 c d \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{63 c^{2} d^{2} \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(9*c*d*(d + e*x)**(5/2)) - 4
*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(63*c**2*d*
*2*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.0779741, size = 65, normalized size = 0.6 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (c d (9 d+7 e x)-2 a e^2\right )}{63 c^2 d^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e^2 + c*d*(9*d + 7*e*x)))
/(63*c^2*d^2*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 69, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -7\,cdex+2\,a{e}^{2}-9\,c{d}^{2} \right ) }{63\,{c}^{2}{d}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/63*(c*d*x+a*e)*(-7*c*d*e*x+2*a*e^2-9*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)
^(5/2)/c^2/d^2/(e*x+d)^(5/2)

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Maxima [A]  time = 0.794018, size = 178, normalized size = 1.63 \[ \frac{2 \,{\left (7 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5} +{\left (9 \, c^{4} d^{5} + 19 \, a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \,{\left (9 \, a c^{3} d^{4} e + 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} +{\left (27 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x\right )} \sqrt{c d x + a e}}{63 \, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*c^4*d^4*e*x^4 + 9*a^3*c*d^2*e^3 - 2*a^4*e^5 + (9*c^4*d^5 + 19*a*c^3*d^3*
e^2)*x^3 + 3*(9*a*c^3*d^4*e + 5*a^2*c^2*d^2*e^3)*x^2 + (27*a^2*c^2*d^3*e^2 + a^3
*c*d*e^4)*x)*sqrt(c*d*x + a*e)/(c^2*d^2)

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Fricas [A]  time = 0.213795, size = 363, normalized size = 3.33 \[ \frac{2 \,{\left (7 \, c^{5} d^{5} e^{2} x^{6} + 9 \, a^{4} c d^{3} e^{4} - 2 \, a^{5} d e^{6} + 2 \,{\left (8 \, c^{5} d^{6} e + 13 \, a c^{4} d^{4} e^{3}\right )} x^{5} +{\left (9 \, c^{5} d^{7} + 62 \, a c^{4} d^{5} e^{2} + 34 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{4} + 4 \,{\left (9 \, a c^{4} d^{6} e + 22 \, a^{2} c^{3} d^{4} e^{3} + 4 \, a^{3} c^{2} d^{2} e^{5}\right )} x^{3} +{\left (54 \, a^{2} c^{3} d^{5} e^{2} + 52 \, a^{3} c^{2} d^{3} e^{4} - a^{4} c d e^{6}\right )} x^{2} + 2 \,{\left (18 \, a^{3} c^{2} d^{4} e^{3} + 4 \, a^{4} c d^{2} e^{5} - a^{5} e^{7}\right )} x\right )}}{63 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*c^5*d^5*e^2*x^6 + 9*a^4*c*d^3*e^4 - 2*a^5*d*e^6 + 2*(8*c^5*d^6*e + 13*a*
c^4*d^4*e^3)*x^5 + (9*c^5*d^7 + 62*a*c^4*d^5*e^2 + 34*a^2*c^3*d^3*e^4)*x^4 + 4*(
9*a*c^4*d^6*e + 22*a^2*c^3*d^4*e^3 + 4*a^3*c^2*d^2*e^5)*x^3 + (54*a^2*c^3*d^5*e^
2 + 52*a^3*c^2*d^3*e^4 - a^4*c*d*e^6)*x^2 + 2*(18*a^3*c^2*d^4*e^3 + 4*a^4*c*d^2*
e^5 - a^5*e^7)*x)/(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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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

[Out]

Timed out