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

Optimal. Leaf size=289 \[ \frac{5 g^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{7/2} d^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 g^2 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3 \sqrt{d+e x}}-\frac{10 g \sqrt{d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 1.25042, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104 \[ \frac{5 g^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{7/2} d^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 g^2 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3 \sqrt{d+e x}}-\frac{10 g \sqrt{d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 118.211, size = 280, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{5}{2}}}{3 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{10 g \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}}}{3 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{5 g^{2} \sqrt{f + g x} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c^{3} d^{3} \sqrt{d + e x}} - \frac{5 g^{\frac{3}{2}} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{c^{\frac{7}{2}} d^{\frac{7}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(d + e*x)**(3/2)*(f + g*x)**(5/2)/(3*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*
d**2))**(3/2)) - 10*g*sqrt(d + e*x)*(f + g*x)**(3/2)/(3*c**2*d**2*sqrt(a*d*e + c
*d*e*x**2 + x*(a*e**2 + c*d**2))) + 5*g**2*sqrt(f + g*x)*sqrt(a*d*e + c*d*e*x**2
+ x*(a*e**2 + c*d**2))/(c**3*d**3*sqrt(d + e*x)) - 5*g**(3/2)*(a*e*g - c*d*f)*s
qrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*atanh(sqrt(c)*sqrt(d)*sqrt(f + g*x
)/(sqrt(g)*sqrt(a*e + c*d*x)))/(c**(7/2)*d**(7/2)*sqrt(d + e*x)*sqrt(a*e + c*d*x
))

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Mathematica [A]  time = 0.549688, size = 202, normalized size = 0.7 \[ \frac{(d+e x)^{5/2} \left (\frac{5 g^{3/2} (a e+c d x)^{5/2} (c d f-a e g) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )}{c^{7/2} d^{7/2}}-\frac{2 \sqrt{f+g x} (a e+c d x) \left (14 g (a e+c d x) (c d f-a e g)+2 (c d f-a e g)^2-3 g^2 (a e+c d x)^2\right )}{3 c^3 d^3}\right )}{2 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(5/2)*((-2*(a*e + c*d*x)*Sqrt[f + g*x]*(2*(c*d*f - a*e*g)^2 + 14*g*(c
*d*f - a*e*g)*(a*e + c*d*x) - 3*g^2*(a*e + c*d*x)^2))/(3*c^3*d^3) + (5*g^(3/2)*(
c*d*f - a*e*g)*(a*e + c*d*x)^(5/2)*Log[a*e*g + 2*Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*
e + c*d*x]*Sqrt[f + g*x] + c*d*(f + 2*g*x)])/(c^(7/2)*d^(7/2))))/(2*((a*e + c*d*
x)*(d + e*x))^(5/2))

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Maple [B]  time = 0.043, size = 652, normalized size = 2.3 \[ -{\frac{1}{6\, \left ( cdx+ae \right ) ^{2}{c}^{3}{d}^{3}} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){x}^{2}a{c}^{2}{d}^{2}e{g}^{3}-15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){x}^{2}{c}^{3}{d}^{3}f{g}^{2}+30\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) x{a}^{2}cd{e}^{2}{g}^{3}-30\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) xa{c}^{2}{d}^{2}ef{g}^{2}+15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{3}{e}^{3}{g}^{3}-15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{2}{e}^{2}{g}^{2}fcd-6\,{x}^{2}{c}^{2}{d}^{2}{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}-40\,{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xaecd\sqrt{dgc}+28\,g\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xf{c}^{2}{d}^{2}\sqrt{dgc}-30\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{a}^{2}{e}^{2}{g}^{2}\sqrt{dgc}+20\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }aefgcd\sqrt{dgc}+4\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{f}^{2}{c}^{2}{d}^{2}\sqrt{dgc} \right ) \sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(15*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/
2))/(d*g*c)^(1/2))*x^2*a*c^2*d^2*e*g^3-15*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+
f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*x^2*c^3*d^3*f*g^2+30*ln(1/2*
(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2
))*x*a^2*c*d*e^2*g^3-30*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1
/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*x*a*c^2*d^2*e*f*g^2+15*ln(1/2*(2*x*c*d*g+a*e*g
+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*a^3*e^3*g^3-1
5*ln(1/2*(2*x*c*d*g+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*
g*c)^(1/2))*a^2*e^2*g^2*f*c*d-6*x^2*c^2*d^2*g^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g
*c)^(1/2)-40*g^2*((g*x+f)*(c*d*x+a*e))^(1/2)*x*a*e*c*d*(d*g*c)^(1/2)+28*g*((g*x+
f)*(c*d*x+a*e))^(1/2)*x*f*c^2*d^2*(d*g*c)^(1/2)-30*((g*x+f)*(c*d*x+a*e))^(1/2)*a
^2*e^2*g^2*(d*g*c)^(1/2)+20*((g*x+f)*(c*d*x+a*e))^(1/2)*a*e*f*g*c*d*(d*g*c)^(1/2
)+4*((g*x+f)*(c*d*x+a*e))^(1/2)*f^2*c^2*d^2*(d*g*c)^(1/2))*(c*d*e*x^2+a*e^2*x+c*
d^2*x+a*d*e)^(1/2)*(g*x+f)^(1/2)/((g*x+f)*(c*d*x+a*e))^(1/2)/(d*g*c)^(1/2)/(c*d*
x+a*e)^2/c^3/d^3/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{5}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^(5/2)*(g*x + f)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)^(5/2), x)

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Fricas [A]  time = 0.964019, 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)*(g*x + f)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(4*(3*c^2*d^2*g^2*x^2 - 2*c^2*d^2*f^2 - 10*a*c*d*e*f*g + 15*a^2*e^2*g^2 -
2*(7*c^2*d^2*f*g - 10*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(g*x + f) - 15*(a^2*c*d^2*e^2*f*g - a^3*d*e^3*g^2 + (c^3*d^3
*e*f*g - 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 - (a*c^2*d^3*
e + 2*a^2*c*d*e^3)*g^2)*x^2 + ((2*a*c^2*d^3*e + a^2*c*d*e^3)*f*g - (2*a^2*c*d^2*
e^2 + a^3*e^4)*g^2)*x)*sqrt(g/(c*d))*log(-(8*c^2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6
*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x
^2 - 4*(2*c^2*d^2*g*x + c^2*d^2*f + a*c*d*e*g)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(g/(c*d)) + (c^2*d^2*e*f^2 + 2*(4*c^2
*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d)))/(c^5*d^5*e
*x^3 + a^2*c^3*d^4*e^2 + (c^5*d^6 + 2*a*c^4*d^4*e^2)*x^2 + (2*a*c^4*d^5*e + a^2*
c^3*d^3*e^3)*x), 1/6*(2*(3*c^2*d^2*g^2*x^2 - 2*c^2*d^2*f^2 - 10*a*c*d*e*f*g + 15
*a^2*e^2*g^2 - 2*(7*c^2*d^2*f*g - 10*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(g*x + f) + 15*(a^2*c*d^2*e^2*f*g - a^3*d*e^3
*g^2 + (c^3*d^3*e*f*g - 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 - (a*c^2*d^3*e + 2*a^2*c*d*e^3)*g^2)*x^2 + ((2*a*c^2*d^3*e + a^2*c*d*e^3)*f*g
- (2*a^2*c*d^2*e^2 + a^3*e^4)*g^2)*x)*sqrt(-g/(c*d))*arctan(2*sqrt(c*d*e*x^2 + a
*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*g/((2*c*d*e*g*x^2 + c*d^2*
f + a*d*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*x)*sqrt(-g/(c*d)))))/(c^5*d^5*e*x^
3 + a^2*c^3*d^4*e^2 + (c^5*d^6 + 2*a*c^4*d^4*e^2)*x^2 + (2*a*c^4*d^5*e + a^2*c^3
*d^3*e^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)**(5/2)/(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 = 1.02622, 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)*(g*x + f)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x