Optimal. Leaf size=211 \[ \frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},-\frac {(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)} \]
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Rubi [A] time = 0.46, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6725, 135, 133} \[ \frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac {f x}{e},-\frac {(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 6725
Rubi steps
\begin {align*} \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx &=\int \left (-\frac {x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {x^m (e+f x)^n}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\int \frac {x^m (e+f x)^n}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\int \frac {x^m (e+f x)^n}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}\\ &=-\frac {\left ((e+f x)^n \left (1+\frac {f x}{e}\right )^{-n}\right ) \int \frac {x^m \left (1+\frac {f x}{e}\right )^n}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\left ((e+f x)^n \left (1+\frac {f x}{e}\right )^{-n}\right ) \int \frac {x^m \left (1+\frac {f x}{e}\right )^n}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\left ((e+f x)^n \left (1+\frac {f x}{e}\right )^{-n}\right ) \int \frac {x^m \left (1+\frac {f x}{e}\right )^n}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}\\ &=\frac {x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {f x}{e},-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}+\frac {x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {f x}{e},\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}+\frac {x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {f x}{e},-\frac {(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}\\ \end {align*}
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Mathematica [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (f x +e \right )^{n}}{b \,x^{3}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^m\,{\left (e+f\,x\right )}^n}{b\,x^3+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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