3.1991 \(\int \frac{(d+e x)^{5/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=114 \[ -\frac{2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{2 (d+e x)^{3/2}}{3 c d} \]

[Out]

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

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Rubi [A]  time = 0.193457, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^2 d^2}+\frac{2 (d+e x)^{3/2}}{3 c d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.6766, size = 100, normalized size = 0.88 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 c d} - \frac{2 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )}{c^{2} d^{2}} + \frac{2 \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**(3/2)/(3*c*d) - 2*sqrt(d + e*x)*(a*e**2 - c*d**2)/(c**2*d**2) + 2*(
a*e**2 - c*d**2)**(3/2)*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2)
)/(c**(5/2)*d**(5/2))

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Mathematica [A]  time = 0.198741, size = 102, normalized size = 0.89 \[ \frac{2 \sqrt{d+e x} \left (c d (4 d+e x)-3 a e^2\right )}{3 c^2 d^2}-\frac{2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(-3*a*e^2 + c*d*(4*d + e*x)))/(3*c^2*d^2) - (2*(c*d^2 - a*e^2)^
(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(5/2)*d^(
5/2))

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Maple [B]  time = 0.01, size = 211, normalized size = 1.9 \[{\frac{2}{3\,cd} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{a{e}^{2}\sqrt{ex+d}}{{c}^{2}{d}^{2}}}+2\,{\frac{\sqrt{ex+d}}{c}}+2\,{\frac{{a}^{2}{e}^{4}}{{c}^{2}{d}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-4\,{\frac{a{e}^{2}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+2\,{\frac{{d}^{2}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(e*x+d)^(3/2)/c/d-2/c^2/d^2*a*e^2*(e*x+d)^(1/2)+2/c*(e*x+d)^(1/2)+2/c^2/d^2/
((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))*a^
2*e^4-4/c/((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))*a*e^2+2*d^2/((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)^(5/2)/(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 = 0.250532, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (c d e x + 4 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt{e x + d}}{3 \, c^{2} d^{2}}, -\frac{2 \,{\left (3 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (c d e x + 4 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt{e x + d}\right )}}{3 \, c^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*(c*d^2 - a*e^2)*sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e
^2 - 2*sqrt(e*x + d)*c*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) + 2*(c*d*e*
x + 4*c*d^2 - 3*a*e^2)*sqrt(e*x + d))/(c^2*d^2), -2/3*(3*(c*d^2 - a*e^2)*sqrt(-(
c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e*x + d)/sqrt(-(c*d^2 - a*e^2)/(c*d))) - (c*d*
e*x + 4*c*d^2 - 3*a*e^2)*sqrt(e*x + d))/(c^2*d^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)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),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)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

Timed out