3.2013 \(\int \frac{(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=176 \[ \frac{15 e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}+\frac{5 e}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{15 \sqrt{c} \sqrt{d} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]

[Out]

(15*e^2)/(4*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]) - 1/(2*(c*d^2 - a*e^2)*(a*e + c*d*x
)^2*Sqrt[d + e*x]) + (5*e)/(4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*Sqrt[d + e*x]) - (
15*Sqrt[c]*Sqrt[d]*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^
2]])/(4*(c*d^2 - a*e^2)^(7/2))

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Rubi [A]  time = 0.32937, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{15 e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}+\frac{5 e}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{15 \sqrt{c} \sqrt{d} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*e^2)/(4*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]) - 1/(2*(c*d^2 - a*e^2)*(a*e + c*d*x
)^2*Sqrt[d + e*x]) + (5*e)/(4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*Sqrt[d + e*x]) - (
15*Sqrt[c]*Sqrt[d]*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^
2]])/(4*(c*d^2 - a*e^2)^(7/2))

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Rubi in Sympy [A]  time = 67.1226, size = 167, normalized size = 0.95 \[ - \frac{15 \sqrt{c} \sqrt{d} e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 \left (a e^{2} - c d^{2}\right )^{\frac{7}{2}}} - \frac{15 c d e \sqrt{d + e x}}{4 \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{3}} - \frac{5 e}{2 \sqrt{d + e x} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{2 \sqrt{d + e x} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(c)*sqrt(d)*e**2*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2
))/(4*(a*e**2 - c*d**2)**(7/2)) - 15*c*d*e*sqrt(d + e*x)/(4*(a*e + c*d*x)*(a*e**
2 - c*d**2)**3) - 5*e/(2*sqrt(d + e*x)*(a*e + c*d*x)*(a*e**2 - c*d**2)**2) + 1/(
2*sqrt(d + e*x)*(a*e + c*d*x)**2*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.482651, size = 157, normalized size = 0.89 \[ \frac{1}{4} \left (\frac{8 a^2 e^4+a c d e^2 (9 d+25 e x)+c^2 d^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{15 \sqrt{c} \sqrt{d} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((8*a^2*e^4 + a*c*d*e^2*(9*d + 25*e*x) + c^2*d^2*(-2*d^2 + 5*d*e*x + 15*e^2*x^2)
)/((c*d^2 - a*e^2)^3*(a*e + c*d*x)^2*Sqrt[d + e*x]) - (15*Sqrt[c]*Sqrt[d]*e^2*Ar
cTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(7/2
))/4

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Maple [A]  time = 0.028, size = 226, normalized size = 1.3 \[ -2\,{\frac{{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ex+d}}}-{\frac{7\,{d}^{2}{e}^{2}{c}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,d{e}^{4}ca}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{9\,{d}^{3}{e}^{2}{c}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,{e}^{2}cd}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e^2/(a*e^2-c*d^2)^3/(e*x+d)^(1/2)-7/4*e^2/(a*e^2-c*d^2)^3*c^2*d^2/(c*d*e*x+a*
e^2)^2*(e*x+d)^(3/2)-9/4*e^4/(a*e^2-c*d^2)^3*c*d/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)
*a+9/4*e^2/(a*e^2-c*d^2)^3*c^2*d^3/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)-15/4*e^2/(a*e
^2-c*d^2)^3*c*d/((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((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246397, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, c^{2} d^{2} e^{2} x^{2} - 4 \, c^{2} d^{4} + 18 \, a c d^{2} e^{2} + 16 \, a^{2} e^{4} + 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 10 \,{\left (c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x}{8 \,{\left (a^{2} c^{3} d^{6} e^{2} - 3 \, a^{3} c^{2} d^{4} e^{4} + 3 \, a^{4} c d^{2} e^{6} - a^{5} e^{8} +{\left (c^{5} d^{8} - 3 \, a c^{4} d^{6} e^{2} + 3 \, a^{2} c^{3} d^{4} e^{4} - a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{4} d^{7} e - 3 \, a^{2} c^{3} d^{5} e^{3} + 3 \, a^{3} c^{2} d^{3} e^{5} - a^{4} c d e^{7}\right )} x\right )} \sqrt{e x + d}}, \frac{15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 9 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} - 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right ) + 5 \,{\left (c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x}{4 \,{\left (a^{2} c^{3} d^{6} e^{2} - 3 \, a^{3} c^{2} d^{4} e^{4} + 3 \, a^{4} c d^{2} e^{6} - a^{5} e^{8} +{\left (c^{5} d^{8} - 3 \, a c^{4} d^{6} e^{2} + 3 \, a^{2} c^{3} d^{4} e^{4} - a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{4} d^{7} e - 3 \, a^{2} c^{3} d^{5} e^{3} + 3 \, a^{3} c^{2} d^{3} e^{5} - a^{4} c d e^{7}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(30*c^2*d^2*e^2*x^2 - 4*c^2*d^4 + 18*a*c*d^2*e^2 + 16*a^2*e^4 + 15*(c^2*d^2
*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^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*(c^2*d^3*e + 5*a*c*d*e^3)*x)/((a^2*c^3*d^6*e^2 - 3*a
^3*c^2*d^4*e^4 + 3*a^4*c*d^2*e^6 - a^5*e^8 + (c^5*d^8 - 3*a*c^4*d^6*e^2 + 3*a^2*
c^3*d^4*e^4 - a^3*c^2*d^2*e^6)*x^2 + 2*(a*c^4*d^7*e - 3*a^2*c^3*d^5*e^3 + 3*a^3*
c^2*d^3*e^5 - a^4*c*d*e^7)*x)*sqrt(e*x + d)), 1/4*(15*c^2*d^2*e^2*x^2 - 2*c^2*d^
4 + 9*a*c*d^2*e^2 + 8*a^2*e^4 - 15*(c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*s
qrt(e*x + d)*sqrt(-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*(c^2*d^3*e + 5*a*c*d*e^3)*x)/((a^2*c^3*d^6*e
^2 - 3*a^3*c^2*d^4*e^4 + 3*a^4*c*d^2*e^6 - a^5*e^8 + (c^5*d^8 - 3*a*c^4*d^6*e^2
+ 3*a^2*c^3*d^4*e^4 - a^3*c^2*d^2*e^6)*x^2 + 2*(a*c^4*d^7*e - 3*a^2*c^3*d^5*e^3
+ 3*a^3*c^2*d^3*e^5 - a^4*c*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((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,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((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Timed out