3.10.14 \(\int \frac {4 b+a x^3}{(b+a x^3) \sqrt [4]{-b-a x^3+c x^4}} \, dx\)

Optimal. Leaf size=69 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a x^3-b+c x^4}}\right )}{\sqrt [4]{c}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a x^3-b+c x^4}}\right )}{\sqrt [4]{c}} \]

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Rubi [F]  time = 1.54, 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 {4 b+a x^3}{\left (b+a x^3\right ) \sqrt [4]{-b-a x^3+c x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*b + a*x^3)/((b + a*x^3)*(-b - a*x^3 + c*x^4)^(1/4)),x]

[Out]

Defer[Int][(-b - a*x^3 + c*x^4)^(-1/4), x] - b^(1/3)*Defer[Int][1/((-b^(1/3) - a^(1/3)*x)*(-b - a*x^3 + c*x^4)
^(1/4)), x] - b^(1/3)*Defer[Int][1/((-b^(1/3) + (-1)^(1/3)*a^(1/3)*x)*(-b - a*x^3 + c*x^4)^(1/4)), x] - b^(1/3
)*Defer[Int][1/((-b^(1/3) - (-1)^(2/3)*a^(1/3)*x)*(-b - a*x^3 + c*x^4)^(1/4)), x]

Rubi steps

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

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Mathematica [F]  time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 b+a x^3}{\left (b+a x^3\right ) \sqrt [4]{-b-a x^3+c x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4*b + a*x^3)/((b + a*x^3)*(-b - a*x^3 + c*x^4)^(1/4)),x]

[Out]

Integrate[(4*b + a*x^3)/((b + a*x^3)*(-b - a*x^3 + c*x^4)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 0.92, size = 69, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b-a x^3+c x^4}}\right )}{\sqrt [4]{c}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b-a x^3+c x^4}}\right )}{\sqrt [4]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(4*b + a*x^3)/((b + a*x^3)*(-b - a*x^3 + c*x^4)^(1/4)),x]

[Out]

(2*ArcTan[(c^(1/4)*x)/(-b - a*x^3 + c*x^4)^(1/4)])/c^(1/4) + (2*ArcTanh[(c^(1/4)*x)/(-b - a*x^3 + c*x^4)^(1/4)
])/c^(1/4)

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fricas [B]  time = 0.58, size = 138, normalized size = 2.00 \begin {gather*} \frac {4 \, \arctan \left (\frac {\frac {x \sqrt {\frac {\sqrt {c} x^{2} + \sqrt {c x^{4} - a x^{3} - b}}{x^{2}}}}{c^{\frac {1}{4}}} - \frac {{\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}}}{c^{\frac {1}{4}}}}{x}\right )}{c^{\frac {1}{4}}} + \frac {\log \left (\frac {c^{\frac {1}{4}} x + {\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{c^{\frac {1}{4}}} - \frac {\log \left (-\frac {c^{\frac {1}{4}} x - {\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{c^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+4*b)/(a*x^3+b)/(c*x^4-a*x^3-b)^(1/4),x, algorithm="fricas")

[Out]

4*arctan((x*sqrt((sqrt(c)*x^2 + sqrt(c*x^4 - a*x^3 - b))/x^2)/c^(1/4) - (c*x^4 - a*x^3 - b)^(1/4)/c^(1/4))/x)/
c^(1/4) + log((c^(1/4)*x + (c*x^4 - a*x^3 - b)^(1/4))/x)/c^(1/4) - log(-(c^(1/4)*x - (c*x^4 - a*x^3 - b)^(1/4)
)/x)/c^(1/4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{3} + 4 \, b}{{\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+4*b)/(a*x^3+b)/(c*x^4-a*x^3-b)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^3 + 4*b)/((c*x^4 - a*x^3 - b)^(1/4)*(a*x^3 + b)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{3}+4 b}{\left (a \,x^{3}+b \right ) \left (c \,x^{4}-a \,x^{3}-b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3+4*b)/(a*x^3+b)/(c*x^4-a*x^3-b)^(1/4),x)

[Out]

int((a*x^3+4*b)/(a*x^3+b)/(c*x^4-a*x^3-b)^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{3} + 4 \, b}{{\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}} {\left (a x^{3} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+4*b)/(a*x^3+b)/(c*x^4-a*x^3-b)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^3 + 4*b)/((c*x^4 - a*x^3 - b)^(1/4)*(a*x^3 + b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x^3+4\,b}{\left (a\,x^3+b\right )\,{\left (c\,x^4-a\,x^3-b\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*b + a*x^3)/((b + a*x^3)*(c*x^4 - a*x^3 - b)^(1/4)),x)

[Out]

int((4*b + a*x^3)/((b + a*x^3)*(c*x^4 - a*x^3 - b)^(1/4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{3} + 4 b}{\left (a x^{3} + b\right ) \sqrt [4]{- a x^{3} - b + c x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**3+4*b)/(a*x**3+b)/(c*x**4-a*x**3-b)**(1/4),x)

[Out]

Integral((a*x**3 + 4*b)/((a*x**3 + b)*(-a*x**3 - b + c*x**4)**(1/4)), x)

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