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

Optimal. Leaf size=235 \[ \frac{5 c d (c d f-a e g)^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{7/2}}-\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{g^3 \sqrt{d+e x}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 1.22366, antiderivative size = 235, 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 \[ \frac{5 c d (c d f-a e g)^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{7/2}}-\frac{5 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{g^3 \sqrt{d+e x}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 109.327, size = 228, normalized size = 0.97 \[ \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 g^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{5 c d \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{g^{3} \sqrt{d + e x}} - \frac{5 c d \left (a e g - c d f\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{g^{\frac{7}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{g \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3*g**2*(d + e*x)**(3/2)
) + 5*c*d*(a*e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(g**3*s
qrt(d + e*x)) - 5*c*d*(a*e*g - c*d*f)**(3/2)*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)))/g**(7/2) - (a*d*e
+ c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(g*(d + e*x)**(5/2)*(f + g*x))

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Mathematica [A]  time = 0.695137, size = 161, normalized size = 0.69 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{-\frac{3 (c d f-a e g)^2}{f+g x}-2 c d (6 c d f-7 a e g)+2 c^2 d^2 g x}{3 g^3 (a e+c d x)^2}-\frac{5 c d (a e g-c d f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{g^{7/2} (a e+c d x)^{5/2}}\right )}{(d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-2*c*d*(6*c*d*f - 7*a*e*g) + 2*c^2*d^2*g*x -
(3*(c*d*f - a*e*g)^2)/(f + g*x))/(3*g^3*(a*e + c*d*x)^2) - (5*c*d*(-(c*d*f) + a*
e*g)^(3/2)*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d*f) + a*e*g]])/(g^(7/2)
*(a*e + c*d*x)^(5/2))))/(d + e*x)^(5/2)

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Maple [B]  time = 0.039, size = 523, normalized size = 2.2 \[ -{\frac{1}{3\,{g}^{3} \left ( gx+f \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{a}^{2}cd{e}^{2}{g}^{3}-30\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xa{c}^{2}{d}^{2}ef{g}^{2}+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{3}{d}^{3}{f}^{2}g+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){a}^{2}cd{e}^{2}f{g}^{2}-30\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) a{c}^{2}{d}^{2}e{f}^{2}g+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{3}{d}^{3}{f}^{3}-2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{2}-14\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xacde{g}^{2}+10\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}x{c}^{2}{d}^{2}fg+3\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}{e}^{2}{g}^{2}-20\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}acdefg+15\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{2}{d}^{2}{f}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^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))*x*a^2*c*d*e^2*g^3-30*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c
*d*f)*g)^(1/2))*x*a*c^2*d^2*e*f*g^2+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f
)*g)^(1/2))*x*c^3*d^3*f^2*g+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/
2))*a^2*c*d*e^2*f*g^2-30*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*
c^2*d^2*e*f^2*g+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*
f^3-2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x^2*c^2*d^2*g^2-14*((a*e*g-c*d*f
)*g)^(1/2)*(c*d*x+a*e)^(1/2)*x*a*c*d*e*g^2+10*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e
)^(1/2)*x*c^2*d^2*f*g+3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^2*g^2-20
*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g+15*((a*e*g-c*d*f)*g)^(1/2
)*(c*d*x+a*e)^(1/2)*c^2*d^2*f^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/g^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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.315764, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^2),x, algorithm="fricas")

[Out]

[1/6*(4*c^3*d^3*e*g^2*x^4 - 30*a*c^2*d^3*e*f^2 + 40*a^2*c*d^2*e^2*f*g - 6*a^3*d*
e^3*g^2 - 4*(5*c^3*d^3*e*f*g - (c^3*d^4 + 8*a*c^2*d^2*e^2)*g^2)*x^3 - 15*(c^2*d^
2*f^2 - a*c*d*e*f*g + (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)*sqrt(-(c*d*f - a*e*g)/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)*sqrt(e*x + d)*g*
sqrt(-(c*d*f - a*e*g)/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^3*d^3*e*f^2 + 10*(c^3*d^4 - a*c^2*d^2*e^2)*f*g - (16*a*
c^2*d^3*e + 11*a^2*c*d*e^3)*g^2)*x^2 - 2*(15*(c^3*d^4 + a*c^2*d^2*e^2)*f^2 - 10*
(a*c^2*d^3*e + 2*a^2*c*d*e^3)*f*g - (11*a^2*c*d^2*e^2 - 3*a^3*e^4)*g^2)*x)/((g^4
*x + f*g^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)), 1/3*(2*c
^3*d^3*e*g^2*x^4 - 15*a*c^2*d^3*e*f^2 + 20*a^2*c*d^2*e^2*f*g - 3*a^3*d*e^3*g^2 -
2*(5*c^3*d^3*e*f*g - (c^3*d^4 + 8*a*c^2*d^2*e^2)*g^2)*x^3 + 15*(c^2*d^2*f^2 - a
*c*d*e*f*g + (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)*sqrt((c*d*f - a*e*g)/g)*arctan(-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)/((c*d*e*g*x^2 + a*d*e*g + (c*d^2
+ a*e^2)*g*x)*sqrt((c*d*f - a*e*g)/g))) - (15*c^3*d^3*e*f^2 + 10*(c^3*d^4 - a*c
^2*d^2*e^2)*f*g - (16*a*c^2*d^3*e + 11*a^2*c*d*e^3)*g^2)*x^2 - (15*(c^3*d^4 + a*
c^2*d^2*e^2)*f^2 - 10*(a*c^2*d^3*e + 2*a^2*c*d*e^3)*f*g - (11*a^2*c*d^2*e^2 - 3*
a^3*e^4)*g^2)*x)/((g^4*x + f*g^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq
rt(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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**2,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^2),x, algorithm="giac")

[Out]

Exception raised: AttributeError