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

Optimal. Leaf size=159 \[ -\frac{\left (a e^2+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4}-\frac{\sqrt{c} d \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^4}+\frac{\sqrt{a+c x^2} \left (2 \left (a e^2+c d^2\right )-c d e x\right )}{2 e^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.483959, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{\left (a e^2+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4}-\frac{\sqrt{c} d \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^4}+\frac{\sqrt{a+c x^2} \left (2 \left (a e^2+c d^2\right )-c d e x\right )}{2 e^3}+\frac{\left (a+c x^2\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.2446, size = 143, normalized size = 0.9 \[ - \frac{\sqrt{c} d \left (3 a e^{2} + 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 e^{4}} + \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 e} + \frac{\sqrt{a + c x^{2}} \left (2 a e^{2} + 2 c d^{2} - c d e x\right )}{2 e^{3}} - \frac{\left (a e^{2} + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.289527, size = 169, normalized size = 1.06 \[ \frac{e \sqrt{a+c x^2} \left (8 a e^2+c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \left (a e^2+c d^2\right )^{3/2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )-3 \sqrt{c} d \left (3 a e^2+2 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+6 \left (a e^2+c d^2\right )^{3/2} \log (d+e x)}{6 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[a + c*x^2]*(8*a*e^2 + c*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 6*(c*d^2 + a*e^
2)^(3/2)*Log[d + e*x] - 3*Sqrt[c]*d*(2*c*d^2 + 3*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a
+ c*x^2]] - 6*(c*d^2 + a*e^2)^(3/2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[
a + c*x^2]])/(6*e^4)

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Maple [B]  time = 0.012, size = 745, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d),x)

[Out]

1/3/e*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-1/2/e^2*c*d*(c*(d/e+
x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/2/e^2*c^(1/2)*d*ln((-c*d/e+c*(
d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a+1/e*(c*
(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a+1/e^3*(c*(d/e+x)^2-2*c*d/e*
(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*c*d^2-1/e^4*c^(3/2)*d^3*ln((-c*d/e+c*(d/e+x))/c
^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))-1/e/((a*e^2+c*d^2)
/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*
(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2-2/e^3/((a*e^
2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2
)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a*c*d^2-
1/e^5/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^
2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+
x))*c^2*d^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.55231, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, c d^{3} + 3 \, a d e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 6 \,{\left (c d^{2} + a e^{2}\right )}^{\frac{3}{2}} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 8 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{12 \, e^{4}}, -\frac{3 \,{\left (2 \, c d^{3} + 3 \, a d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 3 \,{\left (c d^{2} + a e^{2}\right )}^{\frac{3}{2}} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) -{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 8 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{6 \, e^{4}}, \frac{12 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right ) + 3 \,{\left (2 \, c d^{3} + 3 \, a d e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 8 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{12 \, e^{4}}, -\frac{3 \,{\left (2 \, c d^{3} + 3 \, a d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 6 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right ) -{\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 8 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{6 \, e^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(3*(2*c*d^3 + 3*a*d*e^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*
x - a) + 6*(c*d^2 + a*e^2)^(3/2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2
*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*
x^2 + 2*d*e*x + d^2)) + 2*(2*c*e^3*x^2 - 3*c*d*e^2*x + 6*c*d^2*e + 8*a*e^3)*sqrt
(c*x^2 + a))/e^4, -1/6*(3*(2*c*d^3 + 3*a*d*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2
+ a)*sqrt(-c))) - 3*(c*d^2 + a*e^2)^(3/2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2
- (2*c^2*d^2 + a*c*e^2)*x^2 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 +
a))/(e^2*x^2 + 2*d*e*x + d^2)) - (2*c*e^3*x^2 - 3*c*d*e^2*x + 6*c*d^2*e + 8*a*e^
3)*sqrt(c*x^2 + a))/e^4, 1/12*(12*(c*d^2 + a*e^2)*sqrt(-c*d^2 - a*e^2)*arctan((c
*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) + 3*(2*c*d^3 + 3*a*d*e^2)*sq
rt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(2*c*e^3*x^2 - 3*c*d*e
^2*x + 6*c*d^2*e + 8*a*e^3)*sqrt(c*x^2 + a))/e^4, -1/6*(3*(2*c*d^3 + 3*a*d*e^2)*
sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - 6*(c*d^2 + a*e^2)*sqrt(-c*d^2
- a*e^2)*arctan((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) - (2*c*e^3
*x^2 - 3*c*d*e^2*x + 6*c*d^2*e + 8*a*e^3)*sqrt(c*x^2 + a))/e^4]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22351, size = 238, normalized size = 1.5 \[ \frac{1}{2} \,{\left (2 \, c^{\frac{3}{2}} d^{3} + 3 \, a \sqrt{c} d e^{2}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-4\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{1}{6} \, \sqrt{c x^{2} + a}{\left ({\left (2 \, c x e^{\left (-1\right )} - 3 \, c d e^{\left (-2\right )}\right )} x + \frac{2 \,{\left (3 \, c^{2} d^{2} e^{7} + 4 \, a c e^{9}\right )} e^{\left (-10\right )}}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*c^(3/2)*d^3 + 3*a*sqrt(c)*d*e^2)*e^(-4)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 +
a))) + 2*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 +
a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))*e^(-4)/sqrt(-c*d^2 - a*e^2) + 1/6*sqrt(
c*x^2 + a)*((2*c*x*e^(-1) - 3*c*d*e^(-2))*x + 2*(3*c^2*d^2*e^7 + 4*a*c*e^9)*e^(-
10)/c)