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

Optimal. Leaf size=274 \[ -\frac{15 c^2 d^2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 (c d f-a e g)^{7/2}}-\frac{15 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}-\frac{5 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[Out]

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

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Rubi [A]  time = 1.25028, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{15 c^2 d^2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 (c d f-a e g)^{7/2}}-\frac{15 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^3}-\frac{5 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 116.395, size = 265, normalized size = 0.97 \[ - \frac{15 c^{2} d^{2} \sqrt{g} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{4 \left (a e g - c d f\right )^{\frac{7}{2}}} + \frac{15 c d g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 \sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )^{3}} - \frac{5 g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{2 \sqrt{d + e x} \left (f + g x\right )^{2} \left (a e g - c d f\right )^{2}} + \frac{2 \sqrt{d + e x}}{\left (f + g x\right )^{2} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)/(g*x+f)**3/(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(g)*atanh(sqrt(g)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2
))/(sqrt(d + e*x)*sqrt(a*e*g - c*d*f)))/(4*(a*e*g - c*d*f)**(7/2)) + 15*c*d*g*sq
rt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*sqrt(d + e*x)*(f + g*x)*(a*e*g -
c*d*f)**3) - 5*g*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(2*sqrt(d + e*x
)*(f + g*x)**2*(a*e*g - c*d*f)**2) + 2*sqrt(d + e*x)/((f + g*x)**2*(a*e*g - c*d*
f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))

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Mathematica [A]  time = 0.491026, size = 185, normalized size = 0.68 \[ \frac{\sqrt{d+e x} \left (\sqrt{a e g-c d f} \left (-2 a^2 e^2 g^2+a c d e g (9 f+5 g x)+c^2 d^2 \left (8 f^2+25 f g x+15 g^2 x^2\right )\right )-15 c^2 d^2 \sqrt{g} (f+g x)^2 \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )\right )}{4 (f+g x)^2 \sqrt{(d+e x) (a e+c d x)} (a e g-c d f)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(Sqrt[-(c*d*f) + a*e*g]*(-2*a^2*e^2*g^2 + a*c*d*e*g*(9*f + 5*g*x)
+ c^2*d^2*(8*f^2 + 25*f*g*x + 15*g^2*x^2)) - 15*c^2*d^2*Sqrt[g]*Sqrt[a*e + c*d*
x]*(f + g*x)^2*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d*f) + a*e*g]]))/(4*
(-(c*d*f) + a*e*g)^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^2)

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Maple [A]  time = 0.043, size = 379, normalized size = 1.4 \[ -{\frac{1}{ \left ( 4\,cdx+4\,ae \right ) \left ( aeg-cdf \right ) ^{3} \left ( gx+f \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{3}+30\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}x{c}^{2}{d}^{2}f{g}^{2}+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}{c}^{2}{d}^{2}{f}^{2}g-15\,\sqrt{ \left ( aeg-cdf \right ) g}{x}^{2}{c}^{2}{d}^{2}{g}^{2}-5\,\sqrt{ \left ( aeg-cdf \right ) g}xacde{g}^{2}-25\,\sqrt{ \left ( aeg-cdf \right ) g}x{c}^{2}{d}^{2}fg+2\,\sqrt{ \left ( aeg-cdf \right ) g}{a}^{2}{e}^{2}{g}^{2}-9\,\sqrt{ \left ( aeg-cdf \right ) g}acdefg-8\,\sqrt{ \left ( aeg-cdf \right ) g}{c}^{2}{d}^{2}{f}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(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(g*(c*d*x+a*e)^(1/2)/((a
*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*x^2*c^2*d^2*g^3+30*arctanh(g*(c*d*x+a*e)
^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*x*c^2*d^2*f*g^2+15*arctanh(g*(
c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*c^2*d^2*f^2*g-15*((a
*e*g-c*d*f)*g)^(1/2)*x^2*c^2*d^2*g^2-5*((a*e*g-c*d*f)*g)^(1/2)*x*a*c*d*e*g^2-25*
((a*e*g-c*d*f)*g)^(1/2)*x*c^2*d^2*f*g+2*((a*e*g-c*d*f)*g)^(1/2)*a^2*e^2*g^2-9*((
a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*f*g-8*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f^2)/(e*x+d)
^(1/2)/(c*d*x+a*e)/(a*e*g-c*d*f)^3/(g*x+f)^2/((a*e*g-c*d*f)*g)^(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)^(3/2)*(g*x + f)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302045, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(3/2)*(g*x + f)^3),x, algorithm="fricas")

[Out]

[-1/8*(15*(c^3*d^3*e*g^2*x^4 + a*c^2*d^3*e*f^2 + (2*c^3*d^3*e*f*g + (c^3*d^4 + a
*c^2*d^2*e^2)*g^2)*x^3 + (c^3*d^3*e*f^2 + a*c^2*d^3*e*g^2 + 2*(c^3*d^4 + a*c^2*d
^2*e^2)*f*g)*x^2 + (2*a*c^2*d^3*e*f*g + (c^3*d^4 + a*c^2*d^2*e^2)*f^2)*x)*sqrt(-
g/(c*d*f - a*e*g))*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g + 2*sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(-g/(c*d*f - a*e*g)
) - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) + 2*(15*
c^2*d^2*g^2*x^2 + 8*c^2*d^2*f^2 + 9*a*c*d*e*f*g - 2*a^2*e^2*g^2 + 5*(5*c^2*d^2*f
*g + a*c*d*e*g^2)*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^4*e*f^5 - 3*a^2*c^2*d^3*e^2*f^4*g + 3*a^3*c*d^2*e^3*f^3*g^2 - a^4*d*e^4
*f^2*g^3 + (c^4*d^4*e*f^3*g^2 - 3*a*c^3*d^3*e^2*f^2*g^3 + 3*a^2*c^2*d^2*e^3*f*g^
4 - a^3*c*d*e^4*g^5)*x^4 + (2*c^4*d^4*e*f^4*g + (c^4*d^5 - 5*a*c^3*d^3*e^2)*f^3*
g^2 - 3*(a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^2*g^3 + (3*a^2*c^2*d^3*e^2 + a^3*c*d*e
^4)*f*g^4 - (a^3*c*d^2*e^3 + a^4*e^5)*g^5)*x^3 + (c^4*d^4*e*f^5 - a^4*d*e^4*g^5
+ (2*c^4*d^5 - a*c^3*d^3*e^2)*f^4*g - (5*a*c^3*d^4*e + 3*a^2*c^2*d^2*e^3)*f^3*g^
2 + (3*a^2*c^2*d^3*e^2 + 5*a^3*c*d*e^4)*f^2*g^3 + (a^3*c*d^2*e^3 - 2*a^4*e^5)*f*
g^4)*x^2 - (2*a^4*d*e^4*f*g^4 - (c^4*d^5 + a*c^3*d^3*e^2)*f^5 + (a*c^3*d^4*e + 3
*a^2*c^2*d^2*e^3)*f^4*g + 3*(a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^3*g^2 - (5*a^3*c*d
^2*e^3 - a^4*e^5)*f^2*g^3)*x), 1/4*(15*(c^3*d^3*e*g^2*x^4 + a*c^2*d^3*e*f^2 + (2
*c^3*d^3*e*f*g + (c^3*d^4 + a*c^2*d^2*e^2)*g^2)*x^3 + (c^3*d^3*e*f^2 + a*c^2*d^3
*e*g^2 + 2*(c^3*d^4 + a*c^2*d^2*e^2)*f*g)*x^2 + (2*a*c^2*d^3*e*f*g + (c^3*d^4 +
a*c^2*d^2*e^2)*f^2)*x)*sqrt(g/(c*d*f - a*e*g))*arctan(sqrt(e*x + d)/(sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(g/(c*d*f - a*e*g)))) - (15*c^2*d^2*g^2*x^2
+ 8*c^2*d^2*f^2 + 9*a*c*d*e*f*g - 2*a^2*e^2*g^2 + 5*(5*c^2*d^2*f*g + a*c*d*e*g^
2)*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^4*e*f^
5 - 3*a^2*c^2*d^3*e^2*f^4*g + 3*a^3*c*d^2*e^3*f^3*g^2 - a^4*d*e^4*f^2*g^3 + (c^4
*d^4*e*f^3*g^2 - 3*a*c^3*d^3*e^2*f^2*g^3 + 3*a^2*c^2*d^2*e^3*f*g^4 - a^3*c*d*e^4
*g^5)*x^4 + (2*c^4*d^4*e*f^4*g + (c^4*d^5 - 5*a*c^3*d^3*e^2)*f^3*g^2 - 3*(a*c^3*
d^4*e - a^2*c^2*d^2*e^3)*f^2*g^3 + (3*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*f*g^4 - (a^
3*c*d^2*e^3 + a^4*e^5)*g^5)*x^3 + (c^4*d^4*e*f^5 - a^4*d*e^4*g^5 + (2*c^4*d^5 -
a*c^3*d^3*e^2)*f^4*g - (5*a*c^3*d^4*e + 3*a^2*c^2*d^2*e^3)*f^3*g^2 + (3*a^2*c^2*
d^3*e^2 + 5*a^3*c*d*e^4)*f^2*g^3 + (a^3*c*d^2*e^3 - 2*a^4*e^5)*f*g^4)*x^2 - (2*a
^4*d*e^4*f*g^4 - (c^4*d^5 + a*c^3*d^3*e^2)*f^5 + (a*c^3*d^4*e + 3*a^2*c^2*d^2*e^
3)*f^4*g + 3*(a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^3*g^2 - (5*a^3*c*d^2*e^3 - a^4*e^
5)*f^2*g^3)*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)/(g*x+f)**3/(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 [A]  time = 0.90778, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^(3/2)*(g*x + f)^3),x, algorithm="giac")

[Out]

sage0*x