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

Optimal. Leaf size=106 \[ \frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.180301, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 d e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{e \sqrt{a+c x^2} \left (2 \left (c d^2-a e^2\right )+c d e x\right )}{a c^2}-\frac{(d+e x)^2 (a e-c d x)}{a c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.925, size = 99, normalized size = 0.93 \[ \frac{3 d e^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{\left (d + e x\right )^{2} \left (a e - c d x\right )}{a c \sqrt{a + c x^{2}}} + \frac{e \sqrt{a + c x^{2}} \left (4 a e^{2} - 4 c d^{2} - 2 c d e x\right )}{2 a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.159658, size = 91, normalized size = 0.86 \[ \frac{2 a^2 e^3+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )+c^2 d^3 x}{a c^2 \sqrt{a+c x^2}}+\frac{3 d e^2 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 118, normalized size = 1.1 \[{\frac{{d}^{3}x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{e}^{3}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{c{x}^{2}+a}}}-3\,{\frac{d{e}^{2}x}{c\sqrt{c{x}^{2}+a}}}+3\,{\frac{d{e}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }{{c}^{3/2}}}-3\,{\frac{{d}^{2}e}{c\sqrt{c{x}^{2}+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235336, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 3 \,{\left (a c^{2} d e^{2} x^{2} + a^{2} c d e^{2}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{c}}, \frac{{\left (a c e^{3} x^{2} - 3 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 3 \,{\left (a c^{2} d e^{2} x^{2} + a^{2} c d e^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221405, size = 135, normalized size = 1.27 \[ -\frac{3 \, d e^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} + \frac{x{\left (\frac{x e^{3}}{c} + \frac{c^{3} d^{3} - 3 \, a c^{2} d e^{2}}{a c^{3}}\right )} - \frac{3 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}}{a c^{3}}}{\sqrt{c x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*d*e^2*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2) + (x*(x*e^3/c + (c^3*d^3
- 3*a*c^2*d*e^2)/(a*c^3)) - (3*a*c^2*d^2*e - 2*a^2*c*e^3)/(a*c^3))/sqrt(c*x^2 +
a)