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

Optimal. Leaf size=94 \[ -\frac{d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}+\frac{d \sqrt{c+d x^3}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.233898, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}+\frac{d \sqrt{c+d x^3}}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.6761, size = 78, normalized size = 0.83 \[ - \frac{\left (c + d x^{3}\right )^{\frac{3}{2}}}{3 b \left (a + b x^{3}\right )} + \frac{d \sqrt{c + d x^{3}}}{b^{2}} - \frac{d \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.175369, size = 94, normalized size = 1. \[ \frac{1}{3} \sqrt{c+d x^3} \left (\frac{a d-b c}{b^2 \left (a+b x^3\right )}+\frac{2 d}{b^2}\right )-\frac{d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.01, size = 466, normalized size = 5. \[{\frac{ad-bc}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }\sqrt{d{x}^{3}+c}}+{\frac{2\,d}{3\,{b}^{2}}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{2}}\sqrt{2}}{{b}^{2}d}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a*d-b*c)/b^2*(d*x^3+c)^(1/2)/(b*x^3+a)+2/3*d*(d*x^3+c)^(1/2)/b^2+1/2*I/d/b^
2*2^(1/2)*sum((-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^
2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*
3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*
d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1
/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(
2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)
^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(
1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c
*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d
*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b+a))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224473, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b d x^{3} + a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \,{\left (2 \, b d x^{3} - b c + 3 \, a d\right )} \sqrt{d x^{3} + c}}{6 \,{\left (b^{3} x^{3} + a b^{2}\right )}}, -\frac{3 \,{\left (b d x^{3} + a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (2 \, b d x^{3} - b c + 3 \, a d\right )} \sqrt{d x^{3} + c}}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(b*d*x^3 + a*d)*sqrt((b*c - a*d)/b)*log((b*d*x^3 + 2*b*c - a*d - 2*sqrt(
d*x^3 + c)*b*sqrt((b*c - a*d)/b))/(b*x^3 + a)) + 2*(2*b*d*x^3 - b*c + 3*a*d)*sqr
t(d*x^3 + c))/(b^3*x^3 + a*b^2), -1/3*(3*(b*d*x^3 + a*d)*sqrt(-(b*c - a*d)/b)*ar
ctan(sqrt(d*x^3 + c)/sqrt(-(b*c - a*d)/b)) - (2*b*d*x^3 - b*c + 3*a*d)*sqrt(d*x^
3 + c))/(b^3*x^3 + a*b^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(x**2*(d*x**3+c)**(3/2)/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219285, size = 161, normalized size = 1.71 \[ \frac{1}{3} \, d{\left (\frac{3 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} + \frac{2 \, \sqrt{d x^{3} + c}}{b^{2}} - \frac{\sqrt{d x^{3} + c} b c - \sqrt{d x^{3} + c} a d}{{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d*(3*(b*c - a*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c
+ a*b*d)*b^2) + 2*sqrt(d*x^3 + c)/b^2 - (sqrt(d*x^3 + c)*b*c - sqrt(d*x^3 + c)*
a*d)/(((d*x^3 + c)*b - b*c + a*d)*b^2))