3.756 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{7/2}} \, dx\)

Optimal. Leaf size=274 \[ \frac{2 c^{5/2} d^{5/2} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^3 \sqrt{d+e x} \sqrt{f+g x}}-\frac{2 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2} (f+g x)^{3/2}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^{5/2}} \]

[Out]

(-2*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^3*Sqrt[d + e*x]*Sqrt
[f + g*x]) - (2*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)*(f + g*x)^(3/2)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(
5*g*(d + e*x)^(5/2)*(f + g*x)^(5/2)) + (2*c^(5/2)*d^(5/2)*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])])/
(g^(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 = 1.18096, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 c^{5/2} d^{5/2} \sqrt{d+e x} \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g^3 \sqrt{d+e x} \sqrt{f+g x}}-\frac{2 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2} (f+g x)^{3/2}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^{5/2}} \]

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)^(7/2)),x]

[Out]

(-2*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^3*Sqrt[d + e*x]*Sqrt
[f + g*x]) - (2*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)*(f + g*x)^(3/2)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(
5*g*(d + e*x)^(5/2)*(f + g*x)^(5/2)) + (2*c^(5/2)*d^(5/2)*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])])/
(g^(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 = 109.725, size = 267, normalized size = 0.97 \[ \frac{2 c^{\frac{5}{2}} d^{\frac{5}{2}} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a e + c d x}}{\sqrt{c} \sqrt{d} \sqrt{f + g x}} \right )}}{g^{\frac{7}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} - \frac{2 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{g^{3} \sqrt{d + e x} \sqrt{f + g x}} - \frac{2 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 g^{2} \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{3}{2}}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{5 g \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.694894, size = 185, normalized size = 0.68 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{c^{5/2} d^{5/2} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )}{g^{7/2} (a e+c d x)^{5/2}}-\frac{2 \left (-11 c d (f+g x) (c d f-a e g)+3 (c d f-a e g)^2+23 c^2 d^2 (f+g x)^2\right )}{15 g^3 (f+g x)^{5/2} (a e+c d x)^2}\right )}{(d+e x)^{5/2}} \]

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)^(7/2)),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-2*(3*(c*d*f - a*e*g)^2 - 11*c*d*(c*d*f - a*e
*g)*(f + g*x) + 23*c^2*d^2*(f + g*x)^2))/(15*g^3*(a*e + c*d*x)^2*(f + g*x)^(5/2)
) + (c^(5/2)*d^(5/2)*Log[a*e*g + 2*Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x]*Sqr
t[f + g*x] + c*d*(f + 2*g*x)])/(g^(7/2)*(a*e + c*d*x)^(5/2))))/(d + e*x)^(5/2)

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Maple [B]  time = 0.046, size = 511, normalized size = 1.9 \[{\frac{1}{15\,{g}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){x}^{3}{c}^{3}{d}^{3}{g}^{3}+45\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){x}^{2}{c}^{3}{d}^{3}f{g}^{2}+45\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) x{c}^{3}{d}^{3}{f}^{2}g+15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){f}^{3}{c}^{3}{d}^{3}-46\,{x}^{2}{c}^{2}{d}^{2}{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}-22\,{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xaecd\sqrt{dgc}-70\,g\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xf{c}^{2}{d}^{2}\sqrt{dgc}-6\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{a}^{2}{e}^{2}{g}^{2}\sqrt{dgc}-10\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }aefgcd\sqrt{dgc}-30\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{f}^{2}{c}^{2}{d}^{2}\sqrt{dgc} \right ){\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}} \left ( gx+f \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(7/2),x)

[Out]

1/15*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(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))*x^3*c^3*d^3*g^3+45*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+45*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^3*d^3*f^2*g+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))*f^3*c^3*d^3-
46*x^2*c^2*d^2*g^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)-22*g^2*((g*x+f)*(c*
d*x+a*e))^(1/2)*x*a*e*c*d*(d*g*c)^(1/2)-70*g*((g*x+f)*(c*d*x+a*e))^(1/2)*x*f*c^2
*d^2*(d*g*c)^(1/2)-6*((g*x+f)*(c*d*x+a*e))^(1/2)*a^2*e^2*g^2*(d*g*c)^(1/2)-10*((
g*x+f)*(c*d*x+a*e))^(1/2)*a*e*f*g*c*d*(d*g*c)^(1/2)-30*((g*x+f)*(c*d*x+a*e))^(1/
2)*f^2*c^2*d^2*(d*g*c)^(1/2))/((g*x+f)*(c*d*x+a*e))^(1/2)/(d*g*c)^(1/2)/g^3/(g*x
+f)^(5/2)/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{7}{2}}}\,{d x} \]

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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(7/2)),x, algorithm="maxima")

[Out]

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)^(7/2)), x)

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Fricas [A]  time = 0.916216, 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)^(7/2)),x, algorithm="fricas")

[Out]

[-1/30*(4*(23*c^2*d^2*g^2*x^2 + 15*c^2*d^2*f^2 + 5*a*c*d*e*f*g + 3*a^2*e^2*g^2 +
(35*c^2*d^2*f*g + 11*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*(c^2*d^2*e*g^3*x^4 + c^2*d^3*f^3 + (3*c^2*d^2
*e*f*g^2 + c^2*d^3*g^3)*x^3 + 3*(c^2*d^2*e*f^2*g + c^2*d^3*f*g^2)*x^2 + (c^2*d^2
*e*f^3 + 3*c^2*d^3*f^2*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^6*x^4 + d
*f^3*g^3 + (3*e*f*g^5 + d*g^6)*x^3 + 3*(e*f^2*g^4 + d*f*g^5)*x^2 + (e*f^3*g^3 +
3*d*f^2*g^4)*x), -1/15*(2*(23*c^2*d^2*g^2*x^2 + 15*c^2*d^2*f^2 + 5*a*c*d*e*f*g +
3*a^2*e^2*g^2 + (35*c^2*d^2*f*g + 11*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*(c^2*d^2*e*g^3*x^4 + c^2*d^3*
f^3 + (3*c^2*d^2*e*f*g^2 + c^2*d^3*g^3)*x^3 + 3*(c^2*d^2*e*f^2*g + c^2*d^3*f*g^2
)*x^2 + (c^2*d^2*e*f^3 + 3*c^2*d^3*f^2*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^6*x
^4 + d*f^3*g^3 + (3*e*f*g^5 + d*g^6)*x^3 + 3*(e*f^2*g^4 + d*f*g^5)*x^2 + (e*f^3*
g^3 + 3*d*f^2*g^4)*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)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**(7/2),x)

[Out]

Timed out

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

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)^(7/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError