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

Optimal. Leaf size=313 \[ \frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{7/2} d^{7/2} \sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c^3 d^3 \sqrt{d+e x}}+\frac{5 (f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{12 c^2 d^2 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d \sqrt{d+e x}} \]

[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*c^3*d^3*Sqrt[d + e*x]) + (5*(c*d*f - a*e*g)*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(12*c^2*d^2*Sqrt[d + e*x]) + ((f + g*x)^(5/2)*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d*Sqrt[d + e*x]) + (5*(c*d*f - a*e*g
)^3*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])])/(8*c^(7/2)*d^(7/2)*Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2])

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Rubi [A]  time = 1.4674, antiderivative size = 313, 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 \[ \frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{7/2} d^{7/2} \sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c^3 d^3 \sqrt{d+e x}}+\frac{5 (f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{12 c^2 d^2 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[d + e*x]*(f + g*x)^(5/2))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],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*c^3*d^3*Sqrt[d + e*x]) + (5*(c*d*f - a*e*g)*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(12*c^2*d^2*Sqrt[d + e*x]) + ((f + g*x)^(5/2)*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d*Sqrt[d + e*x]) + (5*(c*d*f - a*e*g
)^3*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])])/(8*c^(7/2)*d^(7/2)*Sqrt[g]*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 = 121.9, size = 301, normalized size = 0.96 \[ \frac{\left (f + g x\right )^{\frac{5}{2}} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 c d \sqrt{d + e x}} - \frac{5 \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{12 c^{2} d^{2} \sqrt{d + e x}} + \frac{5 \sqrt{f + g x} \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 c^{3} d^{3} \sqrt{d + e x}} - \frac{5 \left (a e g - c d f\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{f + g x}}{\sqrt{g} \sqrt{a e + c d x}} \right )}}{8 c^{\frac{7}{2}} d^{\frac{7}{2}} \sqrt{g} \sqrt{d + e x} \sqrt{a e + c d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**(5/2)*(e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.386525, size = 205, normalized size = 0.65 \[ \frac{\sqrt{d+e x} \left (\frac{2 \sqrt{f+g x} (a e+c d x) \left (15 a^2 e^2 g^2-10 a c d e g (4 f+g x)+c^2 d^2 \left (33 f^2+26 f g x+8 g^2 x^2\right )\right )}{3 c^3 d^3}+\frac{5 \sqrt{a e+c d x} (c d f-a e g)^3 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{g} \sqrt{f+g x} \sqrt{a e+c d x}+a e g+c d (f+2 g x)\right )}{c^{7/2} d^{7/2} \sqrt{g}}\right )}{16 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((2*(a*e + c*d*x)*Sqrt[f + g*x]*(15*a^2*e^2*g^2 - 10*a*c*d*e*g*(4
*f + g*x) + c^2*d^2*(33*f^2 + 26*f*g*x + 8*g^2*x^2)))/(3*c^3*d^3) + (5*(c*d*f -
a*e*g)^3*Sqrt[a*e + c*d*x]*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)*Sqrt[g])))/(16*Sqrt[(a*e +
c*d*x)*(d + e*x)])

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Maple [A]  time = 0.052, size = 511, normalized size = 1.6 \[ -{\frac{1}{48\,{c}^{3}{d}^{3}}\sqrt{gx+f}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{3}{e}^{3}{g}^{3}-45\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){a}^{2}{e}^{2}{g}^{2}fcd+45\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ) aeg{f}^{2}{c}^{2}{d}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}}{\sqrt{dgc}}} \right ){f}^{3}{c}^{3}{d}^{3}-16\,{x}^{2}{c}^{2}{d}^{2}{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }\sqrt{dgc}+20\,{g}^{2}\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xaecd\sqrt{dgc}-52\,g\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }xf{c}^{2}{d}^{2}\sqrt{dgc}-30\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{a}^{2}{e}^{2}{g}^{2}\sqrt{dgc}+80\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }aefgcd\sqrt{dgc}-66\,\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }{f}^{2}{c}^{2}{d}^{2}\sqrt{dgc} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( gx+f \right ) \left ( cdx+ae \right ) }}}{\frac{1}{\sqrt{dgc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^(5/2)*(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-1/48*(g*x+f)^(1/2)*(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))*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)+20*g^2
*((g*x+f)*(c*d*x+a*e))^(1/2)*x*a*e*c*d*(d*g*c)^(1/2)-52*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)+80*((g*x+f)*(c*d*x+a*e))^(1/2)*a*e*f*g*c*d*(d*g*c)^(1/2)-66*((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)/c^3/d^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(sqrt(e*x + d)*(g*x + f)^(5/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.05696, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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