Optimal. Leaf size=195 \[ \frac {1463 b^6 d \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 d \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{32768 a^{23/4}}+\frac {\sqrt [4]{a x^4+b x^3} \left (-262144 a^7 c x^2+65536 a^6 b c x+327680 a^5 b^2 c+122880 a^5 b^2 d x^8+6144 a^4 b^3 d x^7-7296 a^3 b^4 d x^6+9120 a^2 b^5 d x^5-12540 a b^6 d x^4+21945 b^7 d x^3\right )}{737280 a^5 b^2 x^3} \]
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Rubi [A] time = 0.75, antiderivative size = 341, normalized size of antiderivative = 1.75, number of steps used = 16, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2052, 2016, 2014, 2021, 2024, 2032, 63, 331, 298, 203, 206} \begin {gather*} \frac {1463 b^6 d x^{9/4} (a x+b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{32768 a^{23/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {1463 b^6 d x^{9/4} (a x+b)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{32768 a^{23/4} \left (a x^4+b x^3\right )^{3/4}}+\frac {1463 b^5 d \sqrt [4]{a x^4+b x^3}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{a x^4+b x^3}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{a x^4+b x^3}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{a x^4+b x^3}}{1920 a^2}-\frac {16 a c \left (a x^4+b x^3\right )^{5/4}}{45 b^2 x^5}+\frac {4 c \left (a x^4+b x^3\right )^{5/4}}{9 b x^6}+\frac {b d x^4 \sqrt [4]{a x^4+b x^3}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{a x^4+b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2014
Rule 2016
Rule 2021
Rule 2024
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx &=\int \left (-\frac {c \sqrt [4]{b x^3+a x^4}}{x^4}+d x^4 \sqrt [4]{b x^3+a x^4}\right ) \, dx\\ &=-\left (c \int \frac {\sqrt [4]{b x^3+a x^4}}{x^4} \, dx\right )+d \int x^4 \sqrt [4]{b x^3+a x^4} \, dx\\ &=\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}+\frac {(4 a c) \int \frac {\sqrt [4]{b x^3+a x^4}}{x^3} \, dx}{9 b}+\frac {1}{24} (b d) \int \frac {x^7}{\left (b x^3+a x^4\right )^{3/4}} \, dx\\ &=\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (19 b^2 d\right ) \int \frac {x^6}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{480 a}\\ &=-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {\left (19 b^3 d\right ) \int \frac {x^5}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{512 a^2}\\ &=\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (209 b^4 d\right ) \int \frac {x^4}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{6144 a^3}\\ &=-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {\left (1463 b^5 d\right ) \int \frac {x^3}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{49152 a^4}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d\right ) \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{65536 a^5}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{65536 a^5 \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d x^{9/4} (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 )}{16384 a^5 \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d x^{9/4} (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 )}{16384 a^5 \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d x^{9/4} (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 )}{32768 a^{11/2} \left (b x^3+a x^4\right )^{3/4}}+\frac {\left (1463 b^6 d x^{9/4} (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 )}{32768 a^{11/2} \left (b x^3+a x^4\right )^{3/4}}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {1463 b^6 d x^{9/4} (b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{32768 a^{23/4} \left (b x^3+a x^4\right )^{3/4}}-\frac {1463 b^6 d x^{9/4} (b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{32768 a^{23/4} \left (b x^3+a x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.36, size = 323, normalized size = 1.66 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x+b)} \left (-4 a^{10} c x^2 \sqrt [4]{\frac {a x}{b}+1}+a^9 b c x \sqrt [4]{\frac {a x}{b}+1}+5 a^8 b^2 c \sqrt [4]{\frac {a x}{b}+1}+44 a^2 b^8 d x^2 \sqrt [4]{\frac {a x}{b}+1}-5 b^{10} d \, _2F_1\left (-\frac {33}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+40 b^{10} d \, _2F_1\left (-\frac {29}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )-140 b^{10} d \, _2F_1\left (-\frac {25}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+280 b^{10} d \, _2F_1\left (-\frac {21}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )-350 b^{10} d \, _2F_1\left (-\frac {17}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+280 b^{10} d \, _2F_1\left (-\frac {13}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )-140 b^{10} d \, _2F_1\left (-\frac {9}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+35 b^{10} d \sqrt [4]{\frac {a x}{b}+1}+79 a b^9 d x \sqrt [4]{\frac {a x}{b}+1}\right )}{45 a^8 b^2 x^3 \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.12, size = 195, normalized size = 1.00 \begin {gather*} \frac {1463 b^6 d \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 d \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{32768 a^{23/4}}+\frac {\sqrt [4]{a x^4+b x^3} \left (-262144 a^7 c x^2+65536 a^6 b c x+327680 a^5 b^2 c+122880 a^5 b^2 d x^8+6144 a^4 b^3 d x^7-7296 a^3 b^4 d x^6+9120 a^2 b^5 d x^5-12540 a b^6 d x^4+21945 b^7 d x^3\right )}{737280 a^5 b^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 385, normalized size = 1.97 \begin {gather*} \frac {263340 \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \arctan \left (-\frac {\left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {3}{4}} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{17} b^{6} d - \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {3}{4}} a^{17} x \sqrt {\frac {\sqrt {a x^{4} + b x^{3}} b^{12} d^{2} + \sqrt {\frac {b^{24} d^{4}}{a^{23}}} a^{12} x^{2}}{x^{2}}}}{b^{24} d^{4} x}\right ) - 65835 \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{6} d + \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 65835 \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{6} d - \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 4 \, {\left (122880 \, a^{5} b^{2} d x^{8} + 6144 \, a^{4} b^{3} d x^{7} - 7296 \, a^{3} b^{4} d x^{6} + 9120 \, a^{2} b^{5} d x^{5} - 12540 \, a b^{6} d x^{4} + 21945 \, b^{7} d x^{3} - 262144 \, a^{7} c x^{2} + 65536 \, a^{6} b c x + 327680 \, a^{5} b^{2} c\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{2949120 \, a^{5} b^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.86, size = 359, normalized size = 1.84 \begin {gather*} \frac {\frac {131670 \, \sqrt {2} b^{7} d \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 )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {131670 \, \sqrt {2} b^{7} d \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 )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {65835 \, \sqrt {2} b^{7} d \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^{5}} + \frac {65835 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{7} d \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^{6}} + \frac {24 \, {\left (7315 \, {\left (a + \frac {b}{x}\right )}^{\frac {21}{4}} b^{7} d - 40755 \, {\left (a + \frac {b}{x}\right )}^{\frac {17}{4}} a b^{7} d + 92910 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} a^{2} b^{7} d - 109782 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a^{3} b^{7} d + 69327 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{4} b^{7} d + 21945 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{5} b^{7} d\right )} x^{6}}{a^{5} b^{6}} + \frac {524288 \, {\left (5 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{8} c - 9 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{8} c\right )}}{b^{9}}}{5898240 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}} \left (d \,x^{8}-c \right )}{x^{4}}\, dx \end {gather*}
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 {{\left (d x^{8} - c\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x^{4}}\,{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 = -0.01 \begin {gather*} \int -\frac {\left (c-d\,x^8\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^4} \,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 {\sqrt [4]{x^{3} \left (a x + b\right )} \left (- c + d x^{8}\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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