Optimal. Leaf size=301 \[ \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right )} \]
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Rubi [A] time = 0.76, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2463, 6725, 68} \[ \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d\right )}+\frac {\sqrt [3]{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e (m+1) (m+2) \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2463
Rule 6725
Rubi steps
\begin {align*} \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)}-\frac {(3 b p) \int \frac {x^2 (d+e x)^{1+m}}{a+b x^3} \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)}-\frac {(3 b p) \int \left (\frac {(d+e x)^{1+m}}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(d+e x)^{1+m}}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {(d+e x)^{1+m}}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)}-\frac {\left (\sqrt [3]{b} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e (1+m)}-\frac {\left (\sqrt [3]{b} p\right ) \int \frac {(d+e x)^{1+m}}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e (1+m)}-\frac {\left (\sqrt [3]{b} p\right ) \int \frac {(d+e x)^{1+m}}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e (1+m)}\\ &=\frac {\sqrt [3]{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 239, normalized size = 0.79 \[ \frac {(d+e x)^{m+1} \left (\log \left (c \left (a+b x^3\right )^p\right )-\frac {\sqrt [3]{b} p (d+e x) \left (-\frac {\, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}-\frac {\, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}-\frac {\, _2F_1\left (1,m+2;m+3;\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{m+2}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ), x\right ) \]
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 \[ \int {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.17, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )\, dx \]
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 \[ \frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p}\right )}{e {\left (m + 1\right )}} + \int -\frac {{\left (3 \, b d p x^{2} - {\left (e {\left (m + 1\right )} \log \relax (c) - 3 \, e p\right )} b x^{3} - a e {\left (m + 1\right )} \log \relax (c)\right )} {\left (e x + d\right )}^{m}}{b e {\left (m + 1\right )} x^{3} + a e {\left (m + 1\right )}}\,{d x} \]
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 \[ \int \ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]
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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