Optimal. Leaf size=119 \[ \frac {7 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{a x^8+b x^7}}{\sqrt [4]{a} x^2}\right )}{64 a^{11/4}}+\frac {7 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a x^8+b x^7}}{\sqrt [4]{a} x^2}\right )}{64 a^{11/4}}+\frac {\left (32 a^2 x^2+4 a b x-7 b^2\right ) \sqrt [4]{a x^8+b x^7}}{96 a^2 x} \]
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Rubi [A] time = 0.25, antiderivative size = 196, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2004, 2024, 2032, 63, 331, 298, 203, 206} \begin {gather*} -\frac {7 b^3 x^{21/4} (a x+b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{11/4} \left (a x^8+b x^7\right )^{3/4}}+\frac {7 b^3 x^{21/4} (a x+b)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{11/4} \left (a x^8+b x^7\right )^{3/4}}-\frac {7 b^2 \sqrt [4]{a x^8+b x^7}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{a x^8+b x^7}+\frac {b \sqrt [4]{a x^8+b x^7}}{24 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2004
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [4]{b x^7+a x^8} \, dx &=\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {1}{12} b \int \frac {x^7}{\left (b x^7+a x^8\right )^{3/4}} \, dx\\ &=\frac {b \sqrt [4]{b x^7+a x^8}}{24 a}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}-\frac {\left (7 b^2\right ) \int \frac {x^6}{\left (b x^7+a x^8\right )^{3/4}} \, dx}{96 a}\\ &=\frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3\right ) \int \frac {x^5}{\left (b x^7+a x^8\right )^{3/4}} \, dx}{128 a^2}\\ &=\frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{128 a^2 \left (b x^7+a x^8\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 a^2 \left (b x^7+a x^8\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{32 a^2 \left (b x^7+a x^8\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{5/2} \left (b x^7+a x^8\right )^{3/4}}-\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{5/2} \left (b x^7+a x^8\right )^{3/4}}\\ &=\frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}-\frac {7 b^3 x^{21/4} (b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/4} \left (b x^7+a x^8\right )^{3/4}}+\frac {7 b^3 x^{21/4} (b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/4} \left (b x^7+a x^8\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 47, normalized size = 0.39 \begin {gather*} \frac {4 x \sqrt [4]{x^7 (a x+b)} \, _2F_1\left (-\frac {1}{4},\frac {11}{4};\frac {15}{4};-\frac {a x}{b}\right )}{11 \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.58, size = 119, normalized size = 1.00 \begin {gather*} \frac {\left (-7 b^2+4 a b x+32 a^2 x^2\right ) \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {7 b^3 \tan ^{-1}\left (\frac {\sqrt [4]{b x^7+a x^8}}{\sqrt [4]{a} x^2}\right )}{64 a^{11/4}}+\frac {7 b^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b x^7+a x^8}}{\sqrt [4]{a} x^2}\right )}{64 a^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 265, normalized size = 2.23 \begin {gather*} -\frac {84 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x \arctan \left (-\frac {{\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} a^{8} b^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {3}{4}} - a^{8} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {3}{4}} x^{2} \sqrt {\frac {a^{6} \sqrt {\frac {b^{12}}{a^{11}}} x^{4} + \sqrt {a x^{8} + b x^{7}} b^{6}}{x^{4}}}}{b^{12} x^{2}}\right ) - 21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x \log \left (\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x^{2} + {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} b^{3}\right )}}{x^{2}}\right ) + 21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x \log \left (-\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x^{2} - {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} b^{3}\right )}}{x^{2}}\right ) - 4 \, {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} {\left (32 \, a^{2} x^{2} + 4 \, a b x - 7 \, b^{2}\right )}}{384 \, a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 261, normalized size = 2.19 \begin {gather*} \frac {\frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} b^{4} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{2}} - \frac {8 \, {\left (7 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{4} - 18 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{4} - 21 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{2} b^{4}\right )} x^{3}}{a^{2} b^{3}}}{768 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a \,x^{8}+b \,x^{7}\right )^{\frac {1}{4}}\, dx\]
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 {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 38, normalized size = 0.32 \begin {gather*} \frac {4\,x\,{\left (a\,x^8+b\,x^7\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {11}{4};\ \frac {15}{4};\ -\frac {a\,x}{b}\right )}{11\,{\left (\frac {a\,x}{b}+1\right )}^{1/4}} \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 \sqrt [4]{a x^{8} + b x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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