3.2011 \(\int \frac{(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=144 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^{3/2}}-\frac{e \sqrt{d+e x}}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2} \]

[Out]

-Sqrt[d + e*x]/(2*c*d*(a*e + c*d*x)^2) - (e*Sqrt[d + e*x])/(4*c*d*(c*d^2 - a*e^2
)*(a*e + c*d*x)) + (e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e
^2]])/(4*c^(3/2)*d^(3/2)*(c*d^2 - a*e^2)^(3/2))

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Rubi [A]  time = 0.249422, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^{3/2}}-\frac{e \sqrt{d+e x}}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-Sqrt[d + e*x]/(2*c*d*(a*e + c*d*x)^2) - (e*Sqrt[d + e*x])/(4*c*d*(c*d^2 - a*e^2
)*(a*e + c*d*x)) + (e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e
^2]])/(4*c^(3/2)*d^(3/2)*(c*d^2 - a*e^2)^(3/2))

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Rubi in Sympy [A]  time = 53.2962, size = 119, normalized size = 0.83 \[ \frac{e \sqrt{d + e x}}{4 c d \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )} - \frac{\sqrt{d + e x}}{2 c d \left (a e + c d x\right )^{2}} + \frac{e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 c^{\frac{3}{2}} d^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

e*sqrt(d + e*x)/(4*c*d*(a*e + c*d*x)*(a*e**2 - c*d**2)) - sqrt(d + e*x)/(2*c*d*(
a*e + c*d*x)**2) + e**2*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2)
)/(4*c**(3/2)*d**(3/2)*(a*e**2 - c*d**2)**(3/2))

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Mathematica [A]  time = 0.198843, size = 131, normalized size = 0.91 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{\sqrt{d+e x} \left (a e^2-c d (2 d+e x)\right )}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(Sqrt[d + e*x]*(a*e^2 - c*d*(2*d + e*x)))/(4*c*d*(c*d^2 - a*e^2)*(a*e + c*d*x)^2
) + (e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*c^(3/2
)*d^(3/2)*(c*d^2 - a*e^2)^(3/2))

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Maple [A]  time = 0.021, size = 142, normalized size = 1. \[{\frac{{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}dc}\sqrt{ex+d}}+{\frac{{e}^{2}}{ \left ( 4\,a{e}^{2}-4\,c{d}^{2} \right ) cd}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

1/4*e^2/(c*d*e*x+a*e^2)^2/(a*e^2-c*d^2)*(e*x+d)^(3/2)-1/4*e^2/(c*d*e*x+a*e^2)^2/
d/c*(e*x+d)^(1/2)+1/4*e^2/(a*e^2-c*d^2)/c/d/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d
*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(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((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230002, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} \sqrt{e x + d} -{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} + 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right )}{8 \,{\left (a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4} +{\left (c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{2} + 2 \,{\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} x\right )} \sqrt{c^{2} d^{3} - a c d e^{2}}}, -\frac{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} \sqrt{e x + d} -{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4} +{\left (c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{2} + 2 \,{\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} x\right )} \sqrt{-c^{2} d^{3} + a c d e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(2*sqrt(c^2*d^3 - a*c*d*e^2)*(c*d*e*x + 2*c*d^2 - a*e^2)*sqrt(e*x + d) - (
c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*log((sqrt(c^2*d^3 - a*c*d*e^2)*(c*d*e
*x + 2*c*d^2 - a*e^2) + 2*(c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(c*d*x + a*e)))/(
(a^2*c^2*d^3*e^2 - a^3*c*d*e^4 + (c^4*d^5 - a*c^3*d^3*e^2)*x^2 + 2*(a*c^3*d^4*e
- a^2*c^2*d^2*e^3)*x)*sqrt(c^2*d^3 - a*c*d*e^2)), -1/4*(sqrt(-c^2*d^3 + a*c*d*e^
2)*(c*d*e*x + 2*c*d^2 - a*e^2)*sqrt(e*x + d) - (c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x
+ a^2*e^4)*arctan(-(c*d^2 - a*e^2)/(sqrt(-c^2*d^3 + a*c*d*e^2)*sqrt(e*x + d))))/
((a^2*c^2*d^3*e^2 - a^3*c*d*e^4 + (c^4*d^5 - a*c^3*d^3*e^2)*x^2 + 2*(a*c^3*d^4*e
- a^2*c^2*d^2*e^3)*x)*sqrt(-c^2*d^3 + a*c*d*e^2))]

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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)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [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)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Timed out