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

Optimal. Leaf size=304 \[ -\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 \sqrt{c} \sqrt{d} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 g^3 \sqrt{d+e x}}-\frac{5 \sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}} \]

[Out]

(5*(c*d*f - a*e*g)^2*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(8*g^3*Sqrt[d + e*x]) - (5*(c*d*f - a*e*g)*Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2)^(3/2))/(12*g^2*(d + e*x)^(3/2)) + (Sqrt[f + g*x]*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3*g*(d + e*x)^(5/2)) - (5*(c*d*f - a*e*g)^3*Sq
rt[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])])/(8*Sqrt[c]*Sqrt[d]*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 = 1.39882, antiderivative size = 304, 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 \[ -\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 \sqrt{c} \sqrt{d} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 g^3 \sqrt{d+e x}}-\frac{5 \sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}} \]

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)*Sqrt[f + g*x]),x]

[Out]

(5*(c*d*f - a*e*g)^2*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(8*g^3*Sqrt[d + e*x]) - (5*(c*d*f - a*e*g)*Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2)^(3/2))/(12*g^2*(d + e*x)^(3/2)) + (Sqrt[f + g*x]*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3*g*(d + e*x)^(5/2)) - (5*(c*d*f - a*e*g)^3*Sq
rt[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])])/(8*Sqrt[c]*Sqrt[d]*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 = 118.3, size = 292, normalized size = 0.96 \[ \frac{\sqrt{f + g x} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{3 g \left (d + e x\right )^{\frac{5}{2}}} + \frac{5 \sqrt{f + g x} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{12 g^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{5 \sqrt{f + g x} \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 g^{3} \sqrt{d + e x}} + \frac{5 \left (a e g - c d f\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{8 \sqrt{c} \sqrt{d} g^{\frac{7}{2}} \sqrt{d + e x} \sqrt{a e + c d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((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)**(1/2),x)

[Out]

sqrt(f + g*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(3*g*(d + e*x)**
(5/2)) + 5*sqrt(f + g*x)*(a*e*g - c*d*f)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**
2))**(3/2)/(12*g**2*(d + e*x)**(3/2)) + 5*sqrt(f + g*x)*(a*e*g - c*d*f)**2*sqrt(
a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(8*g**3*sqrt(d + e*x)) + 5*(a*e*g - c*
d*f)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*atanh(sqrt(c)*sqrt(d)*sqr
t(f + g*x)/(sqrt(g)*sqrt(a*e + c*d*x)))/(8*sqrt(c)*sqrt(d)*g**(7/2)*sqrt(d + e*x
)*sqrt(a*e + c*d*x))

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Mathematica [A]  time = 0.475106, size = 205, normalized size = 0.67 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{2 \sqrt{f+g x} \left (33 a^2 e^2 g^2+2 a c d e g (13 g x-20 f)+c^2 d^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )}{3 g^3 (a e+c d x)^2}+\frac{5 (a e g-c d f)^3 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )}{\sqrt{c} \sqrt{d} g^{7/2} (a e+c d x)^{5/2}}\right )}{16 (d+e x)^{5/2}} \]

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)*Sqrt[f + g*x]),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((2*Sqrt[f + g*x]*(33*a^2*e^2*g^2 + 2*a*c*d*e*g
*(-20*f + 13*g*x) + c^2*d^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2)))/(3*g^3*(a*e + c*d*
x)^2) + (5*(-(c*d*f) + a*e*g)^3*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)])/(Sqrt[c]*Sqrt[d]*g^(7/2)*(a*e + c*d*x)
^(5/2))))/(16*(d + e*x)^(5/2))

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Maple [A]  time = 0.038, size = 508, normalized size = 1.7 \[{\frac{1}{48\,{g}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}\sqrt{gx+f} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{3}{e}^{3}{g}^{3}-45\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{2}{e}^{2}{g}^{2}fcd+45\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) aeg{f}^{2}{c}^{2}{d}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){f}^{3}{c}^{3}{d}^{3}+16\,{x}^{2}{c}^{2}{d}^{2}{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}+52\,{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xaecd\sqrt{dgc}-20\,g\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xf{c}^{2}{d}^{2}\sqrt{dgc}+66\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{a}^{2}{e}^{2}{g}^{2}\sqrt{dgc}-80\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }aefgcd\sqrt{dgc}+30\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{f}^{2}{c}^{2}{d}^{2}\sqrt{dgc} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}}} \]

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)^(1/2),x)

[Out]

1/48*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(g*x+f)^(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))*a^3*e^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))*a^2*e^2*g^2*f*c*d+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))*a*e*g*f^2*c^2*d^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))*
f^3*c^3*d^3+16*x^2*c^2*d^2*g^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)+52*g^2*
((g*x+f)*(c*d*x+a*e))^(1/2)*x*a*e*c*d*(d*g*c)^(1/2)-20*g*((g*x+f)*(c*d*x+a*e))^(
1/2)*x*f*c^2*d^2*(d*g*c)^(1/2)+66*((g*x+f)*(c*d*x+a*e))^(1/2)*a^2*e^2*g^2*(d*g*c
)^(1/2)-80*((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))/(e*x+d)^(1/2)/((g*x+f)*(c*d*x+a*e))^(
1/2)/g^3/(d*g*c)^(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)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.0569, 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)*sqrt(g*x + f)),x, algorithm="fricas")

[Out]

[1/96*(4*(8*c^2*d^2*g^2*x^2 + 15*c^2*d^2*f^2 - 40*a*c*d*e*f*g + 33*a^2*e^2*g^2 -
2*(5*c^2*d^2*f*g - 13*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)*sqrt(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) - 15*(c^3*d^4*f^3 - 3*a*c^2*d^3*e*f^2
*g + 3*a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 + (c^3*d^3*e*f^3 - 3*a*c^2*d^2*e^2*f^
2*g + 3*a^2*c*d*e^3*f*g^2 - a^3*e^4*g^3)*x)*log(-(4*(2*c^2*d^2*g^2*x + c^2*d^2*f
*g + a*c*d*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) + (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 + (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)*sqrt(c*d*g))/(e*x + d)
))/((e*g^3*x + d*g^3)*sqrt(c*d*g)), 1/48*(2*(8*c^2*d^2*g^2*x^2 + 15*c^2*d^2*f^2
- 40*a*c*d*e*f*g + 33*a^2*e^2*g^2 - 2*(5*c^2*d^2*f*g - 13*a*c*d*e*g^2)*x)*sqrt(c
*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) -
15*(c^3*d^4*f^3 - 3*a*c^2*d^3*e*f^2*g + 3*a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 +
(c^3*d^3*e*f^3 - 3*a*c^2*d^2*e^2*f^2*g + 3*a^2*c*d*e^3*f*g^2 - a^3*e^4*g^3)*x)*
arctan(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqrt(e*x + d)*
sqrt(g*x + f)/(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)))/((e*g^3*x + d*g^3)*sqrt(-c*d*g))]

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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)**(1/2),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)*sqrt(g*x + f)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError