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

Optimal. Leaf size=214 \[ \frac{2 c^{3/2} d^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^2 \sqrt{d+e x} \sqrt{f+g x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2} (f+g x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.880688, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 c^{3/2} d^{3/2} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^2 \sqrt{d+e x} \sqrt{f+g x}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2} (f+g x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(5/2)),x]

[Out]

(-2*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^2*Sqrt[d + e*x]*Sqrt[f +
g*x]) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)*
(f + g*x)^(3/2)) + (2*c^(3/2)*d^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(S
qrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(g^(5/2)*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 = 84.9134, size = 207, normalized size = 0.97 \[ \frac{2 c^{\frac{3}{2}} d^{\frac{3}{2}} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a e + c d x}}{\sqrt{c} \sqrt{d} \sqrt{f + g x}} \right )}}{g^{\frac{5}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} - \frac{2 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{g^{2} \sqrt{d + e x} \sqrt{f + g x}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 g \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{3}{2}}} \]

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)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.587753, size = 167, normalized size = 0.78 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (3 c^{3/2} d^{3/2} (f+g x)^{3/2} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )-2 \sqrt{g} \sqrt{a e+c d x} (a e g+c d (3 f+4 g x))\right )}{3 g^{5/2} (f+g x)^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.042, size = 331, normalized size = 1.6 \[{\frac{1}{3\,{g}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){x}^{2}{c}^{2}{d}^{2}{g}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) x{c}^{2}{d}^{2}fg+3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){f}^{2}{c}^{2}{d}^{2}-8\,g\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xcd\sqrt{dgc}-2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }aeg\sqrt{dgc}-6\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }fcd\sqrt{dgc} \right ){\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}} \left ( gx+f \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(5/2),x)

[Out]

1/3*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*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^2*d^2*g^2+6*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*c^2*d^2*f*g+3*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))*f^2*c^2*d^2-8*g*((g*x+f)*(c*d*x+a*e))^(1/2)*x*c*d
*(d*g*c)^(1/2)-2*((g*x+f)*(c*d*x+a*e))^(1/2)*a*e*g*(d*g*c)^(1/2)-6*((g*x+f)*(c*d
*x+a*e))^(1/2)*f*c*d*(d*g*c)^(1/2))/(d*g*c)^(1/2)/((g*x+f)*(c*d*x+a*e))^(1/2)/g^
2/(g*x+f)^(3/2)/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(5/2)),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x +
f)^(5/2)), x)

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Fricas [A]  time = 0.909244, size = 1, normalized size = 0. \[ \left [-\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d g x + 3 \, c d f + a e g\right )} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (c d e g^{2} x^{3} + c d^{2} f^{2} +{\left (2 \, c d e f g + c d^{2} g^{2}\right )} x^{2} +{\left (c d e f^{2} + 2 \, c d^{2} f g\right )} x\right )} \sqrt{\frac{c d}{g}} \log \left (-\frac{8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \,{\left (2 \, c d g^{2} x + c d f g + a e g^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} \sqrt{\frac{c d}{g}} + 8 \,{\left (c^{2} d^{2} e f g +{\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} +{\left (c^{2} d^{2} e f^{2} + 2 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g +{\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{6 \,{\left (e g^{4} x^{3} + d f^{2} g^{2} +{\left (2 \, e f g^{3} + d g^{4}\right )} x^{2} +{\left (e f^{2} g^{2} + 2 \, d f g^{3}\right )} x\right )}}, -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (4 \, c d g x + 3 \, c d f + a e g\right )} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (c d e g^{2} x^{3} + c d^{2} f^{2} +{\left (2 \, c d e f g + c d^{2} g^{2}\right )} x^{2} +{\left (c d e f^{2} + 2 \, c d^{2} f g\right )} x\right )} \sqrt{-\frac{c d}{g}} \arctan \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} c d}{{\left (2 \, c d e g x^{2} + c d^{2} f + a d e g +{\left (c d e f +{\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x\right )} \sqrt{-\frac{c d}{g}}}\right )}{3 \,{\left (e g^{4} x^{3} + d f^{2} g^{2} +{\left (2 \, e f g^{3} + d g^{4}\right )} x^{2} +{\left (e f^{2} g^{2} + 2 \, d f g^{3}\right )} x\right )}}\right ] \]

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)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(5/2)),x, algorithm="fricas")

[Out]

[-1/6*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*g*x + 3*c*d*f + a*e*
g)*sqrt(e*x + d)*sqrt(g*x + f) - 3*(c*d*e*g^2*x^3 + c*d^2*f^2 + (2*c*d*e*f*g + c
*d^2*g^2)*x^2 + (c*d*e*f^2 + 2*c*d^2*f*g)*x)*sqrt(c*d/g)*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 + 4*(2*c*d*g^2*x + c*d*f*g +
a*e*g^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)
*sqrt(c*d/g) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + (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))
)/(e*g^4*x^3 + d*f^2*g^2 + (2*e*f*g^3 + d*g^4)*x^2 + (e*f^2*g^2 + 2*d*f*g^3)*x),
-1/3*(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*g*x + 3*c*d*f + a*e*
g)*sqrt(e*x + d)*sqrt(g*x + f) - 3*(c*d*e*g^2*x^3 + c*d^2*f^2 + (2*c*d*e*f*g + c
*d^2*g^2)*x^2 + (c*d*e*f^2 + 2*c*d^2*f*g)*x)*sqrt(-c*d/g)*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)*c*d/((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(-c*d/g))))/(e*g^4*x
^3 + d*f^2*g^2 + (2*e*f*g^3 + d*g^4)*x^2 + (e*f^2*g^2 + 2*d*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(5/2)),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x +
f)^(5/2)), x)