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

Optimal. Leaf size=268 \[ \frac{5 c d g^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{7/2}}+\frac{5 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{10 g \sqrt{d+e x}}{3 (f+g x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

[Out]

(-2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2)) + (10*g*Sqrt[d + e*x])/(3*(c*d*f - a*e*g)^2*(f + g*x)*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*g^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/((c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c*d*g^(3/2)*ArcTan[(S
qrt[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])])/(c*d*f - a*e*g)^(7/2)

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Rubi [A]  time = 1.18414, antiderivative size = 268, 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 \[ \frac{5 c d g^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{7/2}}+\frac{5 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)^3}+\frac{10 g \sqrt{d+e x}}{3 (f+g x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/((f + g*x)^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))/(3*(c*d*f - a*e*g)*(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2)) + (10*g*Sqrt[d + e*x])/(3*(c*d*f - a*e*g)^2*(f + g*x)*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*g^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/((c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c*d*g^(3/2)*ArcTan[(S
qrt[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])])/(c*d*f - a*e*g)^(7/2)

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Rubi in Sympy [A]  time = 115.781, size = 255, normalized size = 0.95 \[ \frac{5 c d g^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{\left (a e g - c d f\right )^{\frac{7}{2}}} - \frac{5 g^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )^{3}} + \frac{10 g \sqrt{d + e x}}{3 \left (f + g x\right ) \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 \left (f + g x\right ) \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*c*d*g**(3/2)*atanh(sqrt(g)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqr
t(d + e*x)*sqrt(a*e*g - c*d*f)))/(a*e*g - c*d*f)**(7/2) - 5*g**2*sqrt(a*d*e + c*
d*e*x**2 + x*(a*e**2 + c*d**2))/(sqrt(d + e*x)*(f + g*x)*(a*e*g - c*d*f)**3) + 1
0*g*sqrt(d + e*x)/(3*(f + g*x)*(a*e*g - c*d*f)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a
*e**2 + c*d**2))) + 2*(d + e*x)**(3/2)/(3*(f + g*x)*(a*e*g - c*d*f)*(a*d*e + c*d
*e*x**2 + x*(a*e**2 + c*d**2))**(3/2))

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Mathematica [A]  time = 0.533796, size = 180, normalized size = 0.67 \[ \frac{(d+e x)^{3/2} \left (\sqrt{a e g-c d f} \left (-3 a^2 e^2 g^2-2 a c d e g (7 f+10 g x)+c^2 d^2 \left (2 f^2-10 f g x-15 g^2 x^2\right )\right )+15 c d g^{3/2} (f+g x) (a e+c d x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )\right )}{3 (f+g x) ((d+e x) (a e+c d x))^{3/2} (a e g-c d f)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(3/2)*(Sqrt[-(c*d*f) + a*e*g]*(-3*a^2*e^2*g^2 - 2*a*c*d*e*g*(7*f + 10
*g*x) + c^2*d^2*(2*f^2 - 10*f*g*x - 15*g^2*x^2)) + 15*c*d*g^(3/2)*(a*e + c*d*x)^
(3/2)*(f + g*x)*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d*f) + a*e*g]]))/(3
*(-(c*d*f) + a*e*g)^(7/2)*((a*e + c*d*x)*(d + e*x))^(3/2)*(f + g*x))

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Maple [A]  time = 0.042, size = 424, normalized size = 1.6 \[{\frac{1}{3\, \left ( cdx+ae \right ) ^{2} \left ( aeg-cdf \right ) ^{3} \left ( gx+f \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{3}+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xacde{g}^{3}\sqrt{cdx+ae}+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}x{c}^{2}{d}^{2}f{g}^{2}+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) acdef{g}^{2}\sqrt{cdx+ae}-15\,\sqrt{ \left ( aeg-cdf \right ) g}{x}^{2}{c}^{2}{d}^{2}{g}^{2}-20\,\sqrt{ \left ( aeg-cdf \right ) g}xacde{g}^{2}-10\,\sqrt{ \left ( aeg-cdf \right ) g}x{c}^{2}{d}^{2}fg-3\,\sqrt{ \left ( aeg-cdf \right ) g}{a}^{2}{e}^{2}{g}^{2}-14\,\sqrt{ \left ( aeg-cdf \right ) g}acdefg+2\,\sqrt{ \left ( aeg-cdf \right ) g}{c}^{2}{d}^{2}{f}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(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+15*arctanh(g*(c*d*x+a*e)^
(1/2)/((a*e*g-c*d*f)*g)^(1/2))*x*a*c*d*e*g^3*(c*d*x+a*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*c^2*d^2*f*g^2+15*arct
anh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e*f*g^2*(c*d*x+a*e)^(1/2)
-15*((a*e*g-c*d*f)*g)^(1/2)*x^2*c^2*d^2*g^2-20*((a*e*g-c*d*f)*g)^(1/2)*x*a*c*d*e
*g^2-10*((a*e*g-c*d*f)*g)^(1/2)*x*c^2*d^2*f*g-3*((a*e*g-c*d*f)*g)^(1/2)*a^2*e^2*
g^2-14*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*f*g+2*((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)^2/(a*e*g-c*d*f)^3/(g*x+f)/((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)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298935, 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)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^2),x, algorithm="fricas")

[Out]

[-1/6*(15*(c^3*d^3*e*g^2*x^4 + a^2*c*d^2*e^2*f*g + (c^3*d^3*e*f*g + (c^3*d^4 + 2
*a*c^2*d^2*e^2)*g^2)*x^3 + ((c^3*d^4 + 2*a*c^2*d^2*e^2)*f*g + (2*a*c^2*d^3*e + a
^2*c*d*e^3)*g^2)*x^2 + (a^2*c*d^2*e^2*g^2 + (2*a*c^2*d^3*e + a^2*c*d*e^3)*f*g)*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 - 2*c^2*d^2*f^2 + 14*a*c*d*e*f*g + 3*a^2*e^2*g^2 + 10*(c
^2*d^2*f*g + 2*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^2*c^3*d^4*e^2*f^4 - 3*a^3*c^2*d^3*e^3*f^3*g + 3*a^4*c*d^2*e^4*f^2*g
^2 - a^5*d*e^5*f*g^3 + (c^5*d^5*e*f^3*g - 3*a*c^4*d^4*e^2*f^2*g^2 + 3*a^2*c^3*d^
3*e^3*f*g^3 - a^3*c^2*d^2*e^4*g^4)*x^4 + (c^5*d^5*e*f^4 + (c^5*d^6 - a*c^4*d^4*e
^2)*f^3*g - 3*(a*c^4*d^5*e + a^2*c^3*d^3*e^3)*f^2*g^2 + (3*a^2*c^3*d^4*e^2 + 5*a
^3*c^2*d^2*e^4)*f*g^3 - (a^3*c^2*d^3*e^3 + 2*a^4*c*d*e^5)*g^4)*x^3 + ((c^5*d^6 +
2*a*c^4*d^4*e^2)*f^4 - (a*c^4*d^5*e + 5*a^2*c^3*d^3*e^3)*f^3*g - 3*(a^2*c^3*d^4
*e^2 - a^3*c^2*d^2*e^4)*f^2*g^2 + (5*a^3*c^2*d^3*e^3 + a^4*c*d*e^5)*f*g^3 - (2*a
^4*c*d^2*e^4 + a^5*e^6)*g^4)*x^2 - (a^5*d*e^5*g^4 - (2*a*c^4*d^5*e + a^2*c^3*d^3
*e^3)*f^4 + (5*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4)*f^3*g - 3*(a^3*c^2*d^3*e^3 +
a^4*c*d*e^5)*f^2*g^2 - (a^4*c*d^2*e^4 - a^5*e^6)*f*g^3)*x), -1/3*(15*(c^3*d^3*e
*g^2*x^4 + a^2*c*d^2*e^2*f*g + (c^3*d^3*e*f*g + (c^3*d^4 + 2*a*c^2*d^2*e^2)*g^2)
*x^3 + ((c^3*d^4 + 2*a*c^2*d^2*e^2)*f*g + (2*a*c^2*d^3*e + a^2*c*d*e^3)*g^2)*x^2
+ (a^2*c*d^2*e^2*g^2 + (2*a*c^2*d^3*e + a^2*c*d*e^3)*f*g)*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 - 2*c^2*d^2*f^2 + 14*a*c*d*e*f*g + 3*a^2
*e^2*g^2 + 10*(c^2*d^2*f*g + 2*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^2*c^3*d^4*e^2*f^4 - 3*a^3*c^2*d^3*e^3*f^3*g + 3*a^4
*c*d^2*e^4*f^2*g^2 - a^5*d*e^5*f*g^3 + (c^5*d^5*e*f^3*g - 3*a*c^4*d^4*e^2*f^2*g^
2 + 3*a^2*c^3*d^3*e^3*f*g^3 - a^3*c^2*d^2*e^4*g^4)*x^4 + (c^5*d^5*e*f^4 + (c^5*d
^6 - a*c^4*d^4*e^2)*f^3*g - 3*(a*c^4*d^5*e + a^2*c^3*d^3*e^3)*f^2*g^2 + (3*a^2*c
^3*d^4*e^2 + 5*a^3*c^2*d^2*e^4)*f*g^3 - (a^3*c^2*d^3*e^3 + 2*a^4*c*d*e^5)*g^4)*x
^3 + ((c^5*d^6 + 2*a*c^4*d^4*e^2)*f^4 - (a*c^4*d^5*e + 5*a^2*c^3*d^3*e^3)*f^3*g
- 3*(a^2*c^3*d^4*e^2 - a^3*c^2*d^2*e^4)*f^2*g^2 + (5*a^3*c^2*d^3*e^3 + a^4*c*d*e
^5)*f*g^3 - (2*a^4*c*d^2*e^4 + a^5*e^6)*g^4)*x^2 - (a^5*d*e^5*g^4 - (2*a*c^4*d^5
*e + a^2*c^3*d^3*e^3)*f^4 + (5*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4)*f^3*g - 3*(a
^3*c^2*d^3*e^3 + a^4*c*d*e^5)*f^2*g^2 - (a^4*c*d^2*e^4 - a^5*e^6)*f*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)**(5/2)/(g*x+f)**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 = 1.16868, 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)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^2),x, algorithm="giac")

[Out]

sage0*x