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

Optimal. Leaf size=246 \[ -\frac{15 c^2 d^2 \sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 g^{7/2}}+\frac{15 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g^3 \sqrt{d+e x}}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2} \]

[Out]

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

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Rubi [A]  time = 1.14144, antiderivative size = 246, 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 \[ -\frac{15 c^2 d^2 \sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 g^{7/2}}+\frac{15 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g^3 \sqrt{d+e x}}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2} \]

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)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 108.741, size = 236, normalized size = 0.96 \[ \frac{15 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 g^{3} \sqrt{d + e x}} - \frac{15 c^{2} d^{2} \sqrt{a e g - c d f} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{4 g^{\frac{7}{2}}} - \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 g^{2} \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{2 g \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{2}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.737715, size = 178, normalized size = 0.72 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{-2 a^2 e^2 g^2-a c d e g (5 f+9 g x)+c^2 d^2 \left (15 f^2+25 f g x+8 g^2 x^2\right )}{g^3 (f+g x)^2 (a e+c d x)^2}-\frac{15 c^2 d^2 \sqrt{a e g-c d f} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{g^{7/2} (a e+c d x)^{5/2}}\right )}{4 (d+e x)^{5/2}} \]

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)^3),x]

[Out]

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

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Maple [B]  time = 0.04, size = 526, normalized size = 2.1 \[ -{\frac{1}{4\,{g}^{3} \left ( gx+f \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}a{c}^{2}{d}^{2}e{g}^{3}-15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{3}{d}^{3}f{g}^{2}+30\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xa{c}^{2}{d}^{2}ef{g}^{2}-30\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{3}{d}^{3}{f}^{2}g+15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) a{c}^{2}{d}^{2}e{f}^{2}g-15\,{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{3}{d}^{3}{f}^{3}-8\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{g}^{2}+9\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xacde{g}^{2}-25\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}x{c}^{2}{d}^{2}fg+2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{a}^{2}{e}^{2}{g}^{2}+5\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}acdefg-15\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}{c}^{2}{d}^{2}{f}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.568585, 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)^3),x, algorithm="fricas")

[Out]

[1/8*(16*c^3*d^3*e*g^2*x^4 + 30*a*c^2*d^3*e*f^2 - 10*a^2*c*d^2*e^2*f*g - 4*a^3*d
*e^3*g^2 + 2*(25*c^3*d^3*e*f*g + (8*c^3*d^4 - a*c^2*d^2*e^2)*g^2)*x^3 + 15*(c^2*
d^2*g^2*x^2 + 2*c^2*d^2*f*g*x + c^2*d^2*f^2)*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 + 5*(5*c^3*d^4 + 4*a*c^2*d^2*e^2)*f*g - (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 + 5*(4*a*c^2*
d^3*e - a^2*c*d*e^3)*f*g - (11*a^2*c*d^2*e^2 + 2*a^3*e^4)*g^2)*x)/((g^5*x^2 + 2*
f*g^4*x + f^2*g^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)), 1
/4*(8*c^3*d^3*e*g^2*x^4 + 15*a*c^2*d^3*e*f^2 - 5*a^2*c*d^2*e^2*f*g - 2*a^3*d*e^3
*g^2 + (25*c^3*d^3*e*f*g + (8*c^3*d^4 - a*c^2*d^2*e^2)*g^2)*x^3 - 15*(c^2*d^2*g^
2*x^2 + 2*c^2*d^2*f*g*x + c^2*d^2*f^2)*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 + 5*(5*c^3*d^4 + 4*a*c^2
*d^2*e^2)*f*g - (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 + 5*(4*a*c^2*d^3*e - a^2*c*d*e^3)*f*g - (11*a^2*c*d^2*e^2 + 2*a^3*e^
4)*g^2)*x)/((g^5*x^2 + 2*f*g^4*x + f^2*g^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*
e^2)*x)*sqrt(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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)^3),x, algorithm="giac")

[Out]

Timed out