3.393 \(\int \frac{x^2}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=26 \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]

[Out]

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

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Rubi [A]  time = 0.176762, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.79665, size = 20, normalized size = 0.77 \[ \frac{2 \log{\left (c \sqrt{a + b x^{3}} + d \right )}}{3 b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

2*log(c*sqrt(a + b*x**3) + d)/(3*b*c)

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Mathematica [A]  time = 0.0160763, size = 26, normalized size = 1. \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.02, size = 455, normalized size = 17.5 \[{\frac{-{\frac{i}{3}}\sqrt{2}}{{b}^{3}d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}b{c}^{2}+a{c}^{2}-{d}^{2} \right ) }{1\sqrt [3]{-a{b}^{2}}\sqrt{{{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}-i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{b \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -3\,\sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( i\sqrt [3]{-a{b}^{2}}\sqrt{3}{\it \_alpha}\,b-i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+2\,{{\it \_alpha}}^{2}{b}^{2}-\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b- \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},-{\frac{{c}^{2}}{2\,b{d}^{2}} \left ( 2\,i\sqrt{3}\sqrt [3]{-a{b}^{2}}{{\it \_alpha}}^{2}b-i\sqrt{3} \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}+i\sqrt{3}ab-3\, \left ( -a{b}^{2} \right ) ^{2/3}{\it \_alpha}-3\,ab \right ) },\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}}}+{\frac{\ln \left ( b{c}^{2}{x}^{3}+a{c}^{2}-{d}^{2} \right ) }{3\,bc}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.690906, size = 30, normalized size = 1.15 \[ \frac{2 \, \log \left (\sqrt{b x^{3} + a} c + d\right )}{3 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*log(sqrt(b*x^3 + a)*c + d)/(b*c)

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Fricas [A]  time = 0.273522, size = 82, normalized size = 3.15 \[ \frac{\log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + \log \left (\sqrt{b x^{3} + a} c + d\right ) - \log \left (\sqrt{b x^{3} + a} c - d\right )}{3 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273732, size = 31, normalized size = 1.19 \[ \frac{2 \,{\rm ln}\left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{3 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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