Optimal. Leaf size=310 \[ \frac {3 c^2 d^2 \left (a+b x^3\right )^{2/3}}{b}+\frac {4 c d^3 x \left (a+b x^3\right )^{2/3}}{3 b}+\frac {c^4 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {4 a c d^3 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3}}+\frac {2 c^3 d x^2 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}+\frac {d^4 x^5 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {8}{3};-\frac {b x^3}{a}\right )}{5 \sqrt [3]{a+b x^3}}-\frac {c^4 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}}+\frac {2 a c d^3 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 b^{4/3}} \]
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Rubi [A]
time = 0.11, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1907, 245,
372, 371, 267, 327} \begin {gather*} -\frac {4 a c d^3 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3}}+\frac {c^4 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {2 a c d^3 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 b^{4/3}}-\frac {c^4 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}+\frac {2 c^3 d x^2 \sqrt [3]{\frac {b x^3}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}+\frac {3 c^2 d^2 \left (a+b x^3\right )^{2/3}}{b}+\frac {4 c d^3 x \left (a+b x^3\right )^{2/3}}{3 b}+\frac {d^4 x^5 \sqrt [3]{\frac {b x^3}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {8}{3};-\frac {b x^3}{a}\right )}{5 \sqrt [3]{a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 267
Rule 327
Rule 371
Rule 372
Rule 1907
Rubi steps
\begin {align*} \int \frac {(c+d x)^4}{\sqrt [3]{a+b x^3}} \, dx &=\int \left (\frac {c^4}{\sqrt [3]{a+b x^3}}+\frac {4 c^3 d x}{\sqrt [3]{a+b x^3}}+\frac {6 c^2 d^2 x^2}{\sqrt [3]{a+b x^3}}+\frac {4 c d^3 x^3}{\sqrt [3]{a+b x^3}}+\frac {d^4 x^4}{\sqrt [3]{a+b x^3}}\right ) \, dx\\ &=c^4 \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx+\left (4 c^3 d\right ) \int \frac {x}{\sqrt [3]{a+b x^3}} \, dx+\left (6 c^2 d^2\right ) \int \frac {x^2}{\sqrt [3]{a+b x^3}} \, dx+\left (4 c d^3\right ) \int \frac {x^3}{\sqrt [3]{a+b x^3}} \, dx+d^4 \int \frac {x^4}{\sqrt [3]{a+b x^3}} \, dx\\ &=\frac {3 c^2 d^2 \left (a+b x^3\right )^{2/3}}{b}+\frac {4 c d^3 x \left (a+b x^3\right )^{2/3}}{3 b}+\frac {c^4 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {c^4 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}}-\frac {\left (4 a c d^3\right ) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{3 b}+\frac {\left (4 c^3 d \sqrt [3]{1+\frac {b x^3}{a}}\right ) \int \frac {x}{\sqrt [3]{1+\frac {b x^3}{a}}} \, dx}{\sqrt [3]{a+b x^3}}+\frac {\left (d^4 \sqrt [3]{1+\frac {b x^3}{a}}\right ) \int \frac {x^4}{\sqrt [3]{1+\frac {b x^3}{a}}} \, dx}{\sqrt [3]{a+b x^3}}\\ &=\frac {3 c^2 d^2 \left (a+b x^3\right )^{2/3}}{b}+\frac {4 c d^3 x \left (a+b x^3\right )^{2/3}}{3 b}+\frac {c^4 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {4 a c d^3 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3}}+\frac {2 c^3 d x^2 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{\sqrt [3]{a+b x^3}}+\frac {d^4 x^5 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {8}{3};-\frac {b x^3}{a}\right )}{5 \sqrt [3]{a+b x^3}}-\frac {c^4 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}}+\frac {2 a c d^3 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 10.35, size = 392, normalized size = 1.26 \begin {gather*} \frac {180 b^{4/3} c^3 d x^2 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )+18 b^{4/3} d^4 x^5 \sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {8}{3};-\frac {b x^3}{a}\right )+5 c \left (54 a \sqrt [3]{b} c d^2+24 a \sqrt [3]{b} d^3 x+54 b^{4/3} c d^2 x^3+24 b^{4/3} d^3 x^4+2 \sqrt {3} \left (3 b c^3-4 a d^3\right ) \sqrt [3]{a+b x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )+2 \left (-3 b c^3+4 a d^3\right ) \sqrt [3]{a+b x^3} \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+3 b c^3 \sqrt [3]{a+b x^3} \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )-4 a d^3 \sqrt [3]{a+b x^3} \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )\right )}{90 b^{4/3} \sqrt [3]{a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{4}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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Sympy [A]
time = 2.59, size = 206, normalized size = 0.66 \begin {gather*} 6 c^{2} d^{2} \left (\begin {cases} \frac {x^{3}}{3 \sqrt [3]{a}} & \text {for}\: b = 0 \\\frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{2 b} & \text {otherwise} \end {cases}\right ) + \frac {c^{4} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {4 c^{3} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} + \frac {4 c d^{3} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {d^{4} x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {8}{3}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^4}{{\left (b\,x^3+a\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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