3.5.2 \(\int \frac {(-2 b+a x^3) \sqrt {b+a x^3}}{x^2 (b-x^2+a x^3)} \, dx\) [402]

Optimal. Leaf size=33 \[ \frac {2 \sqrt {b+a x^3}}{x}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {b+a x^3}}\right ) \]

[Out]

2*(a*x^3+b)^(1/2)/x-2*arctanh(x/(a*x^3+b)^(1/2))

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Rubi [F]
time = 0.83, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^2 \left (b-x^2+a x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[b + a*x^3])/x - (6*a^(1/3)*Sqrt[b + a*x^3])/((1 + Sqrt[3])*b^(1/3) + a^(1/3)*x) + (3*3^(1/4)*Sqrt[2 -
Sqrt[3]]*a^(1/3)*b^(1/3)*(b^(1/3) + a^(1/3)*x)*Sqrt[(b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2)/((1 + Sqrt[3])
*b^(1/3) + a^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*b^(1/3) + a^(1/3)*x)/((1 + Sqrt[3])*b^(1/3) + a^(1/3)
*x)], -7 - 4*Sqrt[3]])/(Sqrt[(b^(1/3)*(b^(1/3) + a^(1/3)*x))/((1 + Sqrt[3])*b^(1/3) + a^(1/3)*x)^2]*Sqrt[b + a
*x^3]) - (2*Sqrt[2]*3^(3/4)*a^(1/3)*b^(1/3)*(b^(1/3) + a^(1/3)*x)*Sqrt[(b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*
x^2)/((1 + Sqrt[3])*b^(1/3) + a^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*b^(1/3) + a^(1/3)*x)/((1 + Sqrt[3]
)*b^(1/3) + a^(1/3)*x)], -7 - 4*Sqrt[3]])/(Sqrt[(b^(1/3)*(b^(1/3) + a^(1/3)*x))/((1 + Sqrt[3])*b^(1/3) + a^(1/
3)*x)^2]*Sqrt[b + a*x^3]) - 2*Defer[Int][Sqrt[b + a*x^3]/(b - x^2 + a*x^3), x] + 3*a*Defer[Int][(x*Sqrt[b + a*
x^3])/(b - x^2 + a*x^3), x]

Rubi steps

\begin {align*} \int \frac {\left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{x^2 \left (b-x^2+a x^3\right )} \, dx &=\int \left (-\frac {2 \sqrt {b+a x^3}}{x^2}+\frac {(-2+3 a x) \sqrt {b+a x^3}}{b-x^2+a x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {b+a x^3}}{x^2} \, dx\right )+\int \frac {(-2+3 a x) \sqrt {b+a x^3}}{b-x^2+a x^3} \, dx\\ &=\frac {2 \sqrt {b+a x^3}}{x}-(3 a) \int \frac {x}{\sqrt {b+a x^3}} \, dx+\int \left (-\frac {2 \sqrt {b+a x^3}}{b-x^2+a x^3}+\frac {3 a x \sqrt {b+a x^3}}{b-x^2+a x^3}\right ) \, dx\\ &=\frac {2 \sqrt {b+a x^3}}{x}-2 \int \frac {\sqrt {b+a x^3}}{b-x^2+a x^3} \, dx-\left (3 a^{2/3}\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}{\sqrt {b+a x^3}} \, dx+(3 a) \int \frac {x \sqrt {b+a x^3}}{b-x^2+a x^3} \, dx-\left (3 \sqrt {2 \left (2-\sqrt {3}\right )} a^{2/3} \sqrt [3]{b}\right ) \int \frac {1}{\sqrt {b+a x^3}} \, dx\\ &=\frac {2 \sqrt {b+a x^3}}{x}-\frac {6 \sqrt [3]{a} \sqrt {b+a x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} \sqrt {b+a x^3}}-\frac {2 \sqrt {2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{b} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b}+\sqrt [3]{a} x\right )^2}} \sqrt {b+a x^3}}-2 \int \frac {\sqrt {b+a x^3}}{b-x^2+a x^3} \, dx+(3 a) \int \frac {x \sqrt {b+a x^3}}{b-x^2+a x^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b+a x^3}}{x}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {b+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.04, size = 841, normalized size = 25.48

method	result	size
default	\(\text {Expression too large to display}\)	\(841\)
risch	\(\text {Expression too large to display}\)	\(841\)
elliptic	\(\text {Expression too large to display}\)	\(841\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3-2*b)*(a*x^3+b)^(1/2)/x^2/(a*x^3-x^2+b),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3-2*b)*(a*x^3+b)^(1/2)/x^2/(a*x^3-x^2+b),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^3 + b)*(a*x^3 - 2*b)/((a*x^3 - x^2 + b)*x^2), x)

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Fricas [A]
time = 14.71, size = 56, normalized size = 1.70 \begin {gather*} \frac {x \log \left (\frac {a x^{3} + x^{2} - 2 \, \sqrt {a x^{3} + b} x + b}{a x^{3} - x^{2} + b}\right ) + 2 \, \sqrt {a x^{3} + b}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3-2*b)*(a*x^3+b)^(1/2)/x^2/(a*x^3-x^2+b),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**3-2*b)*(a*x**3+b)**(1/2)/x**2/(a*x**3-x**2+b),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3-2*b)*(a*x^3+b)^(1/2)/x^2/(a*x^3-x^2+b),x, algorithm="giac")

[Out]

integrate(sqrt(a*x^3 + b)*(a*x^3 - 2*b)/((a*x^3 - x^2 + b)*x^2), x)

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Mupad [B]
time = 0.92, size = 43, normalized size = 1.30 \begin {gather*} \ln \left (\frac {x-\sqrt {a\,x^3+b}}{x+\sqrt {a\,x^3+b}}\right )+\frac {2\,\sqrt {a\,x^3+b}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b + a*x^3)^(1/2)*(2*b - a*x^3))/(x^2*(b + a*x^3 - x^2)),x)

[Out]

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

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