3.2064 \(\int \frac{(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=251 \[ \frac{2 e \sqrt{d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 e}{3 c d \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.503291, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128 \[ \frac{2 e \sqrt{d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 e}{3 c d \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

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)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 103.91, size = 230, normalized size = 0.92 \[ - \frac{2 e^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} + \frac{2 e \sqrt{d + e x}}{\left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 e}{3 c d \sqrt{d + e x} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{2 \sqrt{d + e x}}{3 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.307466, size = 135, normalized size = 0.54 \[ -\frac{2 (d+e x)^{3/2} \left (\sqrt{a e^2-c d^2} \left (c d (d-3 e x)-4 a e^2\right )+3 e^{3/2} (a e+c d x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )\right )}{3 \left (a e^2-c d^2\right )^{5/2} ((d+e x) (a e+c d x))^{3/2}} \]

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)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.03, size = 234, normalized size = 0.9 \[ -{\frac{2}{3\, \left ( cdx+ae \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}xcd{e}^{2}+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{e}^{3}\sqrt{cdx+ae}-3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xcde-4\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}a{e}^{2}+\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}c{d}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(5/2),x)

[Out]

-2/3*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*
e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x*c*d*e^2+3*arctanh(e*(c*d*x+a*e)^(1/2)/(
(a*e^2-c*d^2)*e)^(1/2))*a*e^3*(c*d*x+a*e)^(1/2)-3*((a*e^2-c*d^2)*e)^(1/2)*x*c*d*
e-4*((a*e^2-c*d^2)*e)^(1/2)*a*e^2+((a*e^2-c*d^2)*e)^(1/2)*c*d^2)/(e*x+d)^(1/2)/(
c*d*x+a*e)^2/(a*e^2-c*d^2)^2/((a*e^2-c*d^2)*e)^(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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226985, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (c^{2} d^{2} e^{2} x^{3} + a^{2} d e^{3} +{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{2} +{\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x\right )} \sqrt{-\frac{e}{c d^{2} - a e^{2}}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d e x - c d^{2} + 4 \, a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (a^{2} c^{2} d^{5} e^{2} - 2 \, a^{3} c d^{3} e^{4} + a^{4} d e^{6} +{\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{3} +{\left (c^{4} d^{7} - 3 \, a^{2} c^{2} d^{3} e^{4} + 2 \, a^{3} c d e^{6}\right )} x^{2} +{\left (2 \, a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + a^{4} e^{7}\right )} x\right )}}, -\frac{2 \,{\left (3 \,{\left (c^{2} d^{2} e^{2} x^{3} + a^{2} d e^{3} +{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{2} +{\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x\right )} \sqrt{\frac{e}{c d^{2} - a e^{2}}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d^{2} - a e^{2}}}}\right ) - \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d e x - c d^{2} + 4 \, a e^{2}\right )} \sqrt{e x + d}\right )}}{3 \,{\left (a^{2} c^{2} d^{5} e^{2} - 2 \, a^{3} c d^{3} e^{4} + a^{4} d e^{6} +{\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{3} +{\left (c^{4} d^{7} - 3 \, a^{2} c^{2} d^{3} e^{4} + 2 \, a^{3} c d e^{6}\right )} x^{2} +{\left (2 \, a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + a^{4} e^{7}\right )} x\right )}}\right ] \]

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)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*(c^2*d^2*e^2*x^3 + a^2*d*e^3 + (c^2*d^3*e + 2*a*c*d*e^3)*x^2 + (2*a*c*d^
2*e^2 + a^2*e^4)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d
^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*s
qrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*sqrt(c*d*e
*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*e*x - c*d^2 + 4*a*e^2)*sqrt(e*x + d))/(
a^2*c^2*d^5*e^2 - 2*a^3*c*d^3*e^4 + a^4*d*e^6 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a
^2*c^2*d^2*e^5)*x^3 + (c^4*d^7 - 3*a^2*c^2*d^3*e^4 + 2*a^3*c*d*e^6)*x^2 + (2*a*c
^3*d^6*e - 3*a^2*c^2*d^4*e^3 + a^4*e^7)*x), -2/3*(3*(c^2*d^2*e^2*x^3 + a^2*d*e^3
+ (c^2*d^3*e + 2*a*c*d*e^3)*x^2 + (2*a*c*d^2*e^2 + a^2*e^4)*x)*sqrt(e/(c*d^2 -
a*e^2))*arctan(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e
/(c*d^2 - a*e^2)))) - sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*e*x - c
*d^2 + 4*a*e^2)*sqrt(e*x + d))/(a^2*c^2*d^5*e^2 - 2*a^3*c*d^3*e^4 + a^4*d*e^6 +
(c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^3 + (c^4*d^7 - 3*a^2*c^2*d^3*e
^4 + 2*a^3*c*d*e^6)*x^2 + (2*a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + a^4*e^7)*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)**(3/2)/(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 = 0.640264, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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)^(5/2),x, algorithm="giac")

[Out]

sage0*x