Optimal. Leaf size=78 \[ \frac {a \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}-\frac {\sqrt [4]{a x^3+b}}{3 b x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {266, 51, 63, 212, 206, 203} \begin {gather*} \frac {a \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}-\frac {\sqrt [4]{a x^3+b}}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (b+a x^3\right )^{3/4}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 (b+a x)^{3/4}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{b+a x^3}}{3 b x^3}-\frac {a \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^{3/4}} \, dx,x,x^3\right )}{4 b}\\ &=-\frac {\sqrt [4]{b+a x^3}}{3 b x^3}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^3}\right )}{b}\\ &=-\frac {\sqrt [4]{b+a x^3}}{3 b x^3}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )}{2 b^{3/2}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )}{2 b^{3/2}}\\ &=-\frac {\sqrt [4]{b+a x^3}}{3 b x^3}+\frac {a \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.47 \begin {gather*} \frac {4 a \sqrt [4]{a x^3+b} \, _2F_1\left (\frac {1}{4},2;\frac {5}{4};\frac {a x^3}{b}+1\right )}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 78, normalized size = 1.00 \begin {gather*} \frac {a \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{b}}\right )}{2 b^{7/4}}-\frac {\sqrt [4]{a x^3+b}}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 191, normalized size = 2.45 \begin {gather*} -\frac {12 \, b x^{3} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{3} + b\right )}^{\frac {1}{4}} a b^{5} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {3}{4}} - \sqrt {b^{4} \sqrt {\frac {a^{4}}{b^{7}}} + \sqrt {a x^{3} + b} a^{2}} b^{5} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {3}{4}}}{a^{4}}\right ) - 3 \, b x^{3} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (b^{2} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (a x^{3} + b\right )}^{\frac {1}{4}} a\right ) + 3 \, b x^{3} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-b^{2} \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (a x^{3} + b\right )}^{\frac {1}{4}} a\right ) + 4 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}}}{12 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 221, normalized size = 2.83 \begin {gather*} \frac {\frac {6 \, \sqrt {2} a^{2} \left (-b\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {6 \, \sqrt {2} a^{2} \left (-b\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {3 \, \sqrt {2} a^{2} \left (-b\right )^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a x^{3} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{3} + b} + \sqrt {-b}\right )}{b^{2}} + \frac {3 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{3} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{3} + b} + \sqrt {-b}\right )}{\left (-b\right )^{\frac {3}{4}} b} - \frac {8 \, {\left (a x^{3} + b\right )}^{\frac {1}{4}} a}{b x^{3}}}{24 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (a \,x^{3}+b \right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 94, normalized size = 1.21 \begin {gather*} -\frac {{\left (a x^{3} + b\right )}^{\frac {1}{4}} a}{3 \, {\left ({\left (a x^{3} + b\right )} b - b^{2}\right )}} + \frac {\frac {2 \, a \arctan \left (\frac {{\left (a x^{3} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (\frac {{\left (a x^{3} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{3} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 58, normalized size = 0.74 \begin {gather*} \frac {a\,\mathrm {atan}\left (\frac {{\left (a\,x^3+b\right )}^{1/4}}{b^{1/4}}\right )}{2\,b^{7/4}}-\frac {{\left (a\,x^3+b\right )}^{1/4}}{3\,b\,x^3}+\frac {a\,\mathrm {atanh}\left (\frac {{\left (a\,x^3+b\right )}^{1/4}}{b^{1/4}}\right )}{2\,b^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.20, size = 41, normalized size = 0.53 \begin {gather*} - \frac {\Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{4}} x^{\frac {21}{4}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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