3.2059 \(\int \frac{1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=269 \[ -\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c d}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.511539, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{15 c^2 d^2 \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{15 c^2 d^2 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}}+\frac{5 c d}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 101.664, size = 252, normalized size = 0.94 \[ - \frac{15 c^{2} d^{2} \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{4 \left (a e^{2} - c d^{2}\right )^{\frac{7}{2}}} + \frac{15 c^{2} d^{2} \sqrt{d + e x}}{4 \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{5 c d}{4 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{1}{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.395283, size = 182, normalized size = 0.68 \[ \frac{\sqrt{a e^2-c d^2} \left (-2 a^2 e^4+a c d e^2 (9 d+5 e x)+c^2 d^2 \left (8 d^2+25 d e x+15 e^2 x^2\right )\right )-15 c^2 d^2 \sqrt{e} (d+e x)^2 \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{4 (d+e x)^{3/2} \left (a e^2-c d^2\right )^{7/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.042, size = 384, normalized size = 1.4 \[ -{\frac{1}{ \left ( 4\,cdx+4\,ae \right ) \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{e}^{3}+30\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}x{c}^{2}{d}^{3}{e}^{2}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{c}^{2}{d}^{4}e-15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}{c}^{2}{d}^{2}{e}^{2}-5\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xacd{e}^{3}-25\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{c}^{2}{d}^{3}e+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{2}{e}^{4}-9\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}ac{d}^{2}{e}^{2}-8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-1/4*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*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^2*c^2*d^2*e^3+30*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^2*d^3*e^2+15*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)*c^2*d^4*e-15*((a*e^2-c
*d^2)*e)^(1/2)*x^2*c^2*d^2*e^2-5*((a*e^2-c*d^2)*e)^(1/2)*x*a*c*d*e^3-25*((a*e^2-
c*d^2)*e)^(1/2)*x*c^2*d^3*e+2*((a*e^2-c*d^2)*e)^(1/2)*a^2*e^4-9*((a*e^2-c*d^2)*e
)^(1/2)*a*c*d^2*e^2-8*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(e*x+d)^(5/2)/(c*d*x+a*e)
/(a*e^2-c*d^2)^3/((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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237034, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(3/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/8*(15*(c^3*d^3*e^3*x^4 + a*c^2*d^5*e + (3*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^3 +
3*(c^3*d^5*e + a*c^2*d^3*e^3)*x^2 + (c^3*d^6 + 3*a*c^2*d^4*e^2)*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)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e
^2)))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(15*c^2*d^2*e^2*x^2 + 8*c^2*d^4 + 9*a*c*d^2
*e^2 - 2*a^2*e^4 + 5*(5*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^
2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^3*d^9*e - 3*a^2*c^2*d^7*e^3 + 3*a^3*c*d^5*e^5
- a^4*d^3*e^7 + (c^4*d^7*e^3 - 3*a*c^3*d^5*e^5 + 3*a^2*c^2*d^3*e^7 - a^3*c*d*e^9
)*x^4 + (3*c^4*d^8*e^2 - 8*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 - a^4*e^10)*x^3 + 3
*(c^4*d^9*e - 2*a*c^3*d^7*e^3 + 2*a^3*c*d^3*e^7 - a^4*d*e^9)*x^2 + (c^4*d^10 - 6
*a^2*c^2*d^6*e^4 + 8*a^3*c*d^4*e^6 - 3*a^4*d^2*e^8)*x), 1/4*(15*(c^3*d^3*e^3*x^4
+ a*c^2*d^5*e + (3*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^3 + 3*(c^3*d^5*e + a*c^2*d^3*
e^3)*x^2 + (c^3*d^6 + 3*a*c^2*d^4*e^2)*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)))) -
(15*c^2*d^2*e^2*x^2 + 8*c^2*d^4 + 9*a*c*d^2*e^2 - 2*a^2*e^4 + 5*(5*c^2*d^3*e + a
*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^3*d
^9*e - 3*a^2*c^2*d^7*e^3 + 3*a^3*c*d^5*e^5 - a^4*d^3*e^7 + (c^4*d^7*e^3 - 3*a*c^
3*d^5*e^5 + 3*a^2*c^2*d^3*e^7 - a^3*c*d*e^9)*x^4 + (3*c^4*d^8*e^2 - 8*a*c^3*d^6*
e^4 + 6*a^2*c^2*d^4*e^6 - a^4*e^10)*x^3 + 3*(c^4*d^9*e - 2*a*c^3*d^7*e^3 + 2*a^3
*c*d^3*e^7 - a^4*d*e^9)*x^2 + (c^4*d^10 - 6*a^2*c^2*d^6*e^4 + 8*a^3*c*d^4*e^6 -
3*a^4*d^2*e^8)*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(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \]

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

[Out]

[undef, undef, undef, undef, undef, undef, undef, 2]