Optimal. Leaf size=198 \[ \frac {1463 b^6 c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{32768 a^{23/4}}+\frac {\sqrt [4]{a x^4-b x^3} \left (-262144 a^7 d x^2-65536 a^6 b d x+122880 a^5 b^2 c x^8+327680 a^5 b^2 d-6144 a^4 b^3 c x^7-7296 a^3 b^4 c x^6-9120 a^2 b^5 c x^5-12540 a b^6 c x^4-21945 b^7 c x^3\right )}{737280 a^5 b^2 x^3} \]
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Rubi [A] time = 0.80, antiderivative size = 359, normalized size of antiderivative = 1.81, number of steps used = 16, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2052, 2016, 2014, 2021, 2024, 2032, 63, 331, 298, 203, 206} \begin {gather*} \frac {1463 b^6 c 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 c 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 c \sqrt [4]{a x^4-b x^3}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{a x^4-b x^3}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{a x^4-b x^3}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{a x^4-b x^3}}{1920 a^2}-\frac {16 a d \left (a x^4-b x^3\right )^{5/4}}{45 b^2 x^5}-\frac {b c x^4 \sqrt [4]{a x^4-b x^3}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{a x^4-b x^3}-\frac {4 d \left (a x^4-b x^3\right )^{5/4}}{9 b x^6} \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 (-d+c x^8\right )}{x^4} \, dx &=\int \left (-\frac {d \sqrt [4]{-b x^3+a x^4}}{x^4}+c x^4 \sqrt [4]{-b x^3+a x^4}\right ) \, dx\\ &=c \int x^4 \sqrt [4]{-b x^3+a x^4} \, dx-d \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^4} \, dx\\ &=\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {1}{24} (b c) \int \frac {x^7}{\left (-b x^3+a x^4\right )^{3/4}} \, dx-\frac {(4 a d) \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^3} \, dx}{9 b}\\ &=-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (19 b^2 c\right ) \int \frac {x^6}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{480 a}\\ &=-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (19 b^3 c\right ) \int \frac {x^5}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{512 a^2}\\ &=-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (209 b^4 c\right ) \int \frac {x^4}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{6144 a^3}\\ &=-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^5 c\right ) \int \frac {x^3}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{49152 a^4}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c\right ) \int \frac {x^2}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{65536 a^5}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c 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 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c 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 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c 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 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c 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 c 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 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {1463 b^6 c 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 c 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 = 0.39, size = 325, normalized size = 1.64 \begin {gather*} -\frac {4 \sqrt [4]{x^3 (a x-b)} \left (4 a^{10} d x^2 \sqrt [4]{1-\frac {a x}{b}}+a^9 b d x \sqrt [4]{1-\frac {a x}{b}}-5 a^8 b^2 d \sqrt [4]{1-\frac {a x}{b}}-44 a^2 b^8 c x^2 \sqrt [4]{1-\frac {a x}{b}}+5 b^{10} c \, _2F_1\left (-\frac {33}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-40 b^{10} c \, _2F_1\left (-\frac {29}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )+140 b^{10} c \, _2F_1\left (-\frac {25}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-280 b^{10} c \, _2F_1\left (-\frac {21}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )+350 b^{10} c \, _2F_1\left (-\frac {17}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-280 b^{10} c \, _2F_1\left (-\frac {13}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )+140 b^{10} c \, _2F_1\left (-\frac {9}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-35 b^{10} c \sqrt [4]{1-\frac {a x}{b}}+79 a b^9 c x \sqrt [4]{1-\frac {a x}{b}}\right )}{45 a^8 b^2 x^3 \sqrt [4]{1-\frac {a x}{b}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.39, size = 198, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-b x^3+a x^4} \left (327680 a^5 b^2 d-65536 a^6 b d x-262144 a^7 d x^2-21945 b^7 c x^3-12540 a b^6 c x^4-9120 a^2 b^5 c x^5-7296 a^3 b^4 c x^6-6144 a^4 b^3 c x^7+122880 a^5 b^2 c x^8\right )}{737280 a^5 b^2 x^3}+\frac {1463 b^6 c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{32768 a^{23/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 390, normalized size = 1.97 \begin {gather*} \frac {263340 \, \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \arctan \left (-\frac {\left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {3}{4}} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} a^{17} b^{6} c - \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {3}{4}} a^{17} x \sqrt {\frac {\sqrt {a x^{4} - b x^{3}} b^{12} c^{2} + \sqrt {\frac {b^{24} c^{4}}{a^{23}}} a^{12} x^{2}}{x^{2}}}}{b^{24} c^{4} x}\right ) - 65835 \, \left (\frac {b^{24} c^{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} c + \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 65835 \, \left (\frac {b^{24} c^{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} c - \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 4 \, {\left (122880 \, a^{5} b^{2} c x^{8} - 6144 \, a^{4} b^{3} c x^{7} - 7296 \, a^{3} b^{4} c x^{6} - 9120 \, a^{2} b^{5} c x^{5} - 12540 \, a b^{6} c x^{4} - 21945 \, b^{7} c x^{3} - 262144 \, a^{7} d x^{2} - 65536 \, a^{6} b d x + 327680 \, a^{5} b^{2} d\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 = 0.67, size = 373, normalized size = 1.88 \begin {gather*} \frac {\frac {131670 \, \sqrt {2} b^{7} c \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} c \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} c \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} c \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} c - 40755 \, {\left (a - \frac {b}{x}\right )}^{\frac {17}{4}} a b^{7} c + 92910 \, {\left (a - \frac {b}{x}\right )}^{\frac {13}{4}} a^{2} b^{7} c - 109782 \, {\left (a - \frac {b}{x}\right )}^{\frac {9}{4}} a^{3} b^{7} c + 69327 \, {\left (a - \frac {b}{x}\right )}^{\frac {5}{4}} a^{4} b^{7} c + 21945 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} a^{5} b^{7} c\right )} x^{6}}{a^{5} b^{6}} + \frac {524288 \, {\left (5 \, {\left (a - \frac {b}{x}\right )}^{\frac {9}{4}} b^{8} d - 9 \, {\left (a - \frac {b}{x}\right )}^{\frac {5}{4}} a b^{8} d\right )}}{b^{9}}}{5898240 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}} \left (c \,x^{8}-d \right )}{x^{4}}\, dx\]
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 {{\left (c x^{8} - d\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 (d-c\,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 x^{8} - d\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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