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

Optimal. Leaf size=63 \[ 2 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {a x^3+b x^2+c}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {a x^3+b x^2+c}}\right ) \]

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Rubi [F]  time = 3.01, 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 c-a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+(-3+b) x^2+a x^3\right ) \left (c+(-2+b) x^2+a x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (2 c-a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c+(-3+b) x^2+a x^3\right ) \left (c+(-2+b) x^2+a x^3\right )} \, dx &=\int \frac {\left (2 c-a x^3\right ) \sqrt {c+b x^2+a x^3}}{\left (c-(2-b) x^2+a x^3\right ) \left (c-(3-b) x^2+a x^3\right )} \, dx\\ &=\int \left (\frac {(-4+2 b+3 a x) \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3}+\frac {(6-2 b-3 a x) \sqrt {c+b x^2+a x^3}}{c-(3-b) x^2+a x^3}\right ) \, dx\\ &=\int \frac {(-4+2 b+3 a x) \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3} \, dx+\int \frac {(6-2 b-3 a x) \sqrt {c+b x^2+a x^3}}{c-(3-b) x^2+a x^3} \, dx\\ &=\int \left (\frac {2 \left (1-\frac {3}{b}\right ) b \sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3}+\frac {3 a x \sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3}\right ) \, dx+\int \left (\frac {4 \left (1-\frac {b}{2}\right ) \sqrt {c+b x^2+a x^3}}{-c+(2-b) x^2-a x^3}+\frac {3 a x \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3}\right ) \, dx\\ &=(3 a) \int \frac {x \sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3} \, dx+(3 a) \int \frac {x \sqrt {c+b x^2+a x^3}}{c-(2-b) x^2+a x^3} \, dx+(2 (2-b)) \int \frac {\sqrt {c+b x^2+a x^3}}{-c+(2-b) x^2-a x^3} \, dx-(2 (3-b)) \int \frac {\sqrt {c+b x^2+a x^3}}{-c+(3-b) x^2-a x^3} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.92, size = 21715, normalized size = 344.68 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 2.90, size = 63, normalized size = 1.00 \begin {gather*} -2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {c+b x^2+a x^3}}\right )+2 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {c+b x^2+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.89, size = 293, normalized size = 4.65 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {a^{2} x^{6} + 2 \, {\left (a b + 6 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} + 12 \, b + 4\right )} x^{4} + 2 \, {\left (b + 6\right )} c x^{2} - 4 \, \sqrt {2} {\left (a x^{4} + {\left (b + 2\right )} x^{3} + c x\right )} \sqrt {a x^{3} + b x^{2} + c} + c^{2}}{a^{2} x^{6} + 2 \, {\left (a b - 2 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} - 4 \, b + 4\right )} x^{4} + 2 \, {\left (b - 2\right )} c x^{2} + c^{2}}\right ) + \frac {1}{2} \, \sqrt {3} \log \left (\frac {a^{2} x^{6} + 2 \, {\left (a b + 9 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} + 18 \, b + 9\right )} x^{4} + 2 \, {\left (b + 9\right )} c x^{2} + 4 \, \sqrt {3} {\left (a x^{4} + {\left (b + 3\right )} x^{3} + c x\right )} \sqrt {a x^{3} + b x^{2} + c} + c^{2}}{a^{2} x^{6} + 2 \, {\left (a b - 3 \, a\right )} x^{5} + 2 \, a c x^{3} + {\left (b^{2} - 6 \, b + 9\right )} x^{4} + 2 \, {\left (b - 3\right )} c x^{2} + c^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log((a^2*x^6 + 2*(a*b + 6*a)*x^5 + 2*a*c*x^3 + (b^2 + 12*b + 4)*x^4 + 2*(b + 6)*c*x^2 - 4*sqrt(2)*
(a*x^4 + (b + 2)*x^3 + c*x)*sqrt(a*x^3 + b*x^2 + c) + c^2)/(a^2*x^6 + 2*(a*b - 2*a)*x^5 + 2*a*c*x^3 + (b^2 - 4
*b + 4)*x^4 + 2*(b - 2)*c*x^2 + c^2)) + 1/2*sqrt(3)*log((a^2*x^6 + 2*(a*b + 9*a)*x^5 + 2*a*c*x^3 + (b^2 + 18*b
+ 9)*x^4 + 2*(b + 9)*c*x^2 + 4*sqrt(3)*(a*x^4 + (b + 3)*x^3 + c*x)*sqrt(a*x^3 + b*x^2 + c) + c^2)/(a^2*x^6 +
2*(a*b - 3*a)*x^5 + 2*a*c*x^3 + (b^2 - 6*b + 9)*x^4 + 2*(b - 3)*c*x^2 + c^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {\sqrt {a x^{3} + b x^{2} + c} {\left (a x^{3} - 2 \, c\right )}}{{\left (a x^{3} + {\left (b - 2\right )} x^{2} + c\right )} {\left (a x^{3} + {\left (b - 3\right )} x^{2} + c\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 7.40, size = 16170, normalized size = 256.67

method	result	size
default	\(\text {Expression too large to display}\)	\(16170\)
elliptic	\(\text {Expression too large to display}\)	\(977321\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {\sqrt {a x^{3} + b x^{2} + c} {\left (a x^{3} - 2 \, c\right )}}{{\left (a x^{3} + {\left (b - 2\right )} x^{2} + c\right )} {\left (a x^{3} + {\left (b - 3\right )} x^{2} + c\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 34.98, size = 125, normalized size = 1.98 \begin {gather*} \sqrt {2}\,\ln \left (\frac {c+a\,x^3+b\,x^2+2\,x^2-2\,\sqrt {2}\,x\,\sqrt {a\,x^3+b\,x^2+c}}{c+a\,x^3+b\,x^2-2\,x^2}\right )+\sqrt {3}\,\ln \left (\frac {c+a\,x^3+b\,x^2+3\,x^2+2\,\sqrt {3}\,x\,\sqrt {a\,x^3+b\,x^2+c}}{c+a\,x^3+b\,x^2-3\,x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  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*c)*(a*x**3+b*x**2+c)**(1/2)/(c+(-3+b)*x**2+a*x**3)/(c+(-2+b)*x**2+a*x**3),x)

[Out]

Timed out

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