3.1996 \(\int \frac{1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx\)

Optimal. Leaf size=153 \[ -\frac{2 c^{5/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}+\frac{2 c^2 d^2}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3}+\frac{2 c d}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{2}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \]

[Out]

2/(5*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (2*c*d)/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(
3/2)) + (2*c^2*d^2)/((c*d^2 - a*e^2)^3*Sqrt[d + e*x]) - (2*c^(5/2)*d^(5/2)*ArcTa
nh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(7/2)

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Rubi [A]  time = 0.335626, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 c^{5/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}+\frac{2 c^2 d^2}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3}+\frac{2 c d}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{2}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

2/(5*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (2*c*d)/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(
3/2)) + (2*c^2*d^2)/((c*d^2 - a*e^2)^3*Sqrt[d + e*x]) - (2*c^(5/2)*d^(5/2)*ArcTa
nh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(7/2)

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Rubi in Sympy [A]  time = 62.8147, size = 136, normalized size = 0.89 \[ - \frac{2 c^{\frac{5}{2}} d^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{7}{2}}} - \frac{2 c^{2} d^{2}}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{2 c d}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2}{5 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

-2*c**(5/2)*d**(5/2)*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2))/(
a*e**2 - c*d**2)**(7/2) - 2*c**2*d**2/(sqrt(d + e*x)*(a*e**2 - c*d**2)**3) + 2*c
*d/(3*(d + e*x)**(3/2)*(a*e**2 - c*d**2)**2) - 2/(5*(d + e*x)**(5/2)*(a*e**2 - c
*d**2))

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Mathematica [A]  time = 0.321529, size = 145, normalized size = 0.95 \[ \frac{2 \left (\frac{-3 a^2 e^4+a c d e^2 (11 d+5 e x)-c^2 d^2 \left (23 d^2+35 d e x+15 e^2 x^2\right )}{(d+e x)^{5/2}}+\frac{15 c^{5/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c d^2-a e^2}}\right )}{15 \left (a e^2-c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

(2*((-3*a^2*e^4 + a*c*d*e^2*(11*d + 5*e*x) - c^2*d^2*(23*d^2 + 35*d*e*x + 15*e^2
*x^2))/(d + e*x)^(5/2) + (15*c^(5/2)*d^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e
*x])/Sqrt[c*d^2 - a*e^2]])/Sqrt[c*d^2 - a*e^2]))/(15*(-(c*d^2) + a*e^2)^3)

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Maple [A]  time = 0.017, size = 146, normalized size = 1. \[ -{\frac{2}{5\,a{e}^{2}-5\,c{d}^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-2\,{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ex+d}}}+{\frac{2\,cd}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{{c}^{3}{d}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

-2/5/(a*e^2-c*d^2)/(e*x+d)^(5/2)-2*c^2*d^2/(a*e^2-c*d^2)^3/(e*x+d)^(1/2)+2/3*c*d
/(a*e^2-c*d^2)^2/(e*x+d)^(3/2)-2*c^3*d^3/(a*e^2-c*d^2)^3/((a*e^2-c*d^2)*c*d)^(1/
2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231092, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, c^{2} d^{2} e^{2} x^{2} + 46 \, c^{2} d^{4} - 22 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} - 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} + 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 10 \,{\left (7 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{15 \,{\left (c^{3} d^{8} - 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} - a^{3} d^{2} e^{6} +{\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e - 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} - a^{3} d e^{7}\right )} x\right )} \sqrt{e x + d}}, \frac{2 \,{\left (15 \, c^{2} d^{2} e^{2} x^{2} + 23 \, c^{2} d^{4} - 11 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} - 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right ) + 5 \,{\left (7 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )}}{15 \,{\left (c^{3} d^{8} - 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} - a^{3} d^{2} e^{6} +{\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e - 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} - a^{3} d e^{7}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/15*(30*c^2*d^2*e^2*x^2 + 46*c^2*d^4 - 22*a*c*d^2*e^2 + 6*a^2*e^4 - 15*(c^2*d^
2*e^2*x^2 + 2*c^2*d^3*e*x + c^2*d^4)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 - a*e^2))*log
((c*d*e*x + 2*c*d^2 - a*e^2 + 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 -
a*e^2)))/(c*d*x + a*e)) + 10*(7*c^2*d^3*e - a*c*d*e^3)*x)/((c^3*d^8 - 3*a*c^2*d^
6*e^2 + 3*a^2*c*d^4*e^4 - a^3*d^2*e^6 + (c^3*d^6*e^2 - 3*a*c^2*d^4*e^4 + 3*a^2*c
*d^2*e^6 - a^3*e^8)*x^2 + 2*(c^3*d^7*e - 3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 - a^3
*d*e^7)*x)*sqrt(e*x + d)), 2/15*(15*c^2*d^2*e^2*x^2 + 23*c^2*d^4 - 11*a*c*d^2*e^
2 + 3*a^2*e^4 - 15*(c^2*d^2*e^2*x^2 + 2*c^2*d^3*e*x + c^2*d^4)*sqrt(e*x + d)*sqr
t(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(-c*d/(c*d^2 - a*e^2))/(sqrt
(e*x + d)*c*d)) + 5*(7*c^2*d^3*e - a*c*d*e^3)*x)/((c^3*d^8 - 3*a*c^2*d^6*e^2 + 3
*a^2*c*d^4*e^4 - a^3*d^2*e^6 + (c^3*d^6*e^2 - 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6
- a^3*e^8)*x^2 + 2*(c^3*d^7*e - 3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 - a^3*d*e^7)*x
)*sqrt(e*x + d))]

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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(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Timed out