Optimal. Leaf size=208 \[ \frac {4 \left (a x^4+b x\right )^{3/4}}{9 b x^3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a} b}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a} b} \]
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Rubi [A] time = 1.15, antiderivative size = 343, normalized size of antiderivative = 1.65, number of steps used = 17, number of rules used = 13, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2056, 6725, 264, 329, 275, 240, 212, 206, 203, 466, 465, 494, 453} \begin {gather*} \frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}-\frac {2^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4+b x}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}-\frac {2^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4+b x}}+\frac {4 (a+b) \left (a x^3+b\right )}{9 a b x^2 \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^3+b\right )}{9 a x^2 \sqrt [4]{a x^4+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 264
Rule 275
Rule 329
Rule 453
Rule 465
Rule 466
Rule 494
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {b+a x^6}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {b+a x^6}{x^{13/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \left (\frac {b}{a x^{13/4} \sqrt [4]{b+a x^3}}+\frac {1}{\sqrt [4]{x} \sqrt [4]{b+a x^3}}+\frac {a b+b^2}{a x^{13/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}}\right ) \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}+\frac {\left (b \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}}+\frac {\left (b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \left (-b+a x^{12}\right ) \sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1-a x^4}{x^4 \left (-b+2 a b x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )}{9 a b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-b+2 a b x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )}{9 a b x^2 \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}-\frac {\left (2 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b \sqrt [4]{b x+a x^4}}-\frac {\left (2 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )}{9 a b x^2 \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{b x+a x^4}}\\ \end {align*}
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Mathematica [C] time = 3.24, size = 188, normalized size = 0.90 \begin {gather*} \frac {4 \left (\frac {4 \Gamma \left (\frac {5}{4}\right ) (a+b) \left (a x^3+b\right ) \left (5 \left (-4 a^2 x^6+3 a b x^3+b^2\right ) \, _2F_1\left (1,1;\frac {5}{4};-\frac {2 a x^3}{b-a x^3}\right )-32 a x^3 \left (a x^3+b\right ) \, _2F_1\left (2,2;\frac {9}{4};-\frac {2 a x^3}{b-a x^3}\right )\right )}{a b \Gamma \left (\frac {1}{4}\right ) \left (b-a x^3\right )^2}+15 x^3 \sqrt [4]{\frac {a x^3}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {a x^3}{b}\right )-\frac {5 \left (a x^3+b\right )}{a}\right )}{45 x^2 \sqrt [4]{x \left (a x^3+b\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.98, size = 208, normalized size = 1.00 \begin {gather*} \frac {4 \left (b x+a x^4\right )^{3/4}}{9 b x^3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a} b}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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giac [B] time = 0.21, size = 437, normalized size = 2.10 \begin {gather*} \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a} - \frac {2^{\frac {1}{4}} {\left (a + b\right )} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} + \frac {2^{\frac {1}{4}} {\left (a + b\right )} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, a b} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, a b} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}}}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}+b}{x^{3} \left (a \,x^{3}-b \right ) \left (a \,x^{4}+b x \right )^{\frac {1}{4}}}\, dx\]
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*} \int \frac {a x^{6} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} - b\right )} x^{3}}\,{d x} \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 {a\,x^6+b}{x^3\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} + b}{x^{3} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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