Optimal. Leaf size=221 \[ \frac {\sqrt {2} (a c+b d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x \sqrt [4]{x^4-x^3} \sqrt [4]{c-d}}{x^2 \sqrt {c-d}-\sqrt {d} \sqrt {x^4-x^3}}\right )}{d^{7/4} \sqrt [4]{c-d}}-\frac {\sqrt {2} (a c+b d) \tanh ^{-1}\left (\frac {x^2 \sqrt {c-d}+\sqrt {d} \sqrt {x^4-x^3}}{\sqrt {2} \sqrt [4]{d} x \sqrt [4]{x^4-x^3} \sqrt [4]{c-d}}\right )}{d^{7/4} \sqrt [4]{c-d}}-\frac {4 a \left (x^4-x^3\right )^{3/4}}{3 d x^3} \]
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Rubi [A] time = 0.34, antiderivative size = 179, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2056, 155, 12, 93, 212, 208, 205} \begin {gather*} -\frac {2 \sqrt [4]{x-1} x^{3/4} (a c+b d) \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{d-c}}{\sqrt [4]{d} \sqrt [4]{x-1}}\right )}{d^{7/4} \sqrt [4]{x^4-x^3} \sqrt [4]{d-c}}-\frac {2 \sqrt [4]{x-1} x^{3/4} (a c+b d) \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{d-c}}{\sqrt [4]{d} \sqrt [4]{x-1}}\right )}{d^{7/4} \sqrt [4]{x^4-x^3} \sqrt [4]{d-c}}+\frac {4 a (1-x)}{3 d \sqrt [4]{x^4-x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 155
Rule 205
Rule 208
Rule 212
Rule 2056
Rubi steps
\begin {align*} \int \frac {a+b x}{x (-d+c x) \sqrt [4]{-x^3+x^4}} \, dx &=\frac {\left (\sqrt [4]{-1+x} x^{3/4}\right ) \int \frac {a+b x}{\sqrt [4]{-1+x} x^{7/4} (-d+c x)} \, dx}{\sqrt [4]{-x^3+x^4}}\\ &=\frac {4 a (1-x)}{3 d \sqrt [4]{-x^3+x^4}}-\frac {\left (4 \sqrt [4]{-1+x} x^{3/4}\right ) \int -\frac {3 (a c+b d)}{4 \sqrt [4]{-1+x} x^{3/4} (-d+c x)} \, dx}{3 d \sqrt [4]{-x^3+x^4}}\\ &=\frac {4 a (1-x)}{3 d \sqrt [4]{-x^3+x^4}}+\frac {\left ((a c+b d) \sqrt [4]{-1+x} x^{3/4}\right ) \int \frac {1}{\sqrt [4]{-1+x} x^{3/4} (-d+c x)} \, dx}{d \sqrt [4]{-x^3+x^4}}\\ &=\frac {4 a (1-x)}{3 d \sqrt [4]{-x^3+x^4}}+\frac {\left (4 (a c+b d) \sqrt [4]{-1+x} x^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{-d-(c-d) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{d \sqrt [4]{-x^3+x^4}}\\ &=\frac {4 a (1-x)}{3 d \sqrt [4]{-x^3+x^4}}-\frac {\left (2 (a c+b d) \sqrt [4]{-1+x} x^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d}-\sqrt {-c+d} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{d^{3/2} \sqrt [4]{-x^3+x^4}}-\frac {\left (2 (a c+b d) \sqrt [4]{-1+x} x^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d}+\sqrt {-c+d} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{d^{3/2} \sqrt [4]{-x^3+x^4}}\\ &=\frac {4 a (1-x)}{3 d \sqrt [4]{-x^3+x^4}}-\frac {2 (a c+b d) \sqrt [4]{-1+x} x^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-c+d} \sqrt [4]{x}}{\sqrt [4]{d} \sqrt [4]{-1+x}}\right )}{d^{7/4} \sqrt [4]{-c+d} \sqrt [4]{-x^3+x^4}}-\frac {2 (a c+b d) \sqrt [4]{-1+x} x^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-c+d} \sqrt [4]{x}}{\sqrt [4]{d} \sqrt [4]{-1+x}}\right )}{d^{7/4} \sqrt [4]{-c+d} \sqrt [4]{-x^3+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 69, normalized size = 0.31 \begin {gather*} -\frac {4 \left ((x-1) x^3\right )^{3/4} \left ((a c+b d) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {d-d x}{c x-d x}\right )+a (d-c)\right )}{3 d x^3 (d-c)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.14, size = 221, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} (a c+b d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x \sqrt [4]{x^4-x^3} \sqrt [4]{c-d}}{x^2 \sqrt {c-d}-\sqrt {d} \sqrt {x^4-x^3}}\right )}{d^{7/4} \sqrt [4]{c-d}}-\frac {\sqrt {2} (a c+b d) \tanh ^{-1}\left (\frac {x^2 \sqrt {c-d}+\sqrt {d} \sqrt {x^4-x^3}}{\sqrt {2} \sqrt [4]{d} x \sqrt [4]{x^4-x^3} \sqrt [4]{c-d}}\right )}{d^{7/4} \sqrt [4]{c-d}}-\frac {4 a \left (x^4-x^3\right )^{3/4}}{3 d x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.54, size = 1021, normalized size = 4.62 \begin {gather*} -\frac {12 \, d x^{3} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \arctan \left (\frac {d^{2} x \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \sqrt {-\frac {{\left (a^{4} c^{5} d^{3} - b^{4} d^{8} - {\left (a^{4} - 4 \, a^{3} b\right )} c^{4} d^{4} - 2 \, {\left (2 \, a^{3} b - 3 \, a^{2} b^{2}\right )} c^{3} d^{5} - 2 \, {\left (3 \, a^{2} b^{2} - 2 \, a b^{3}\right )} c^{2} d^{6} - {\left (4 \, a b^{3} - b^{4}\right )} c d^{7}\right )} x^{2} \sqrt {-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}} - {\left (a^{6} c^{6} + 6 \, a^{5} b c^{5} d + 15 \, a^{4} b^{2} c^{4} d^{2} + 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{2} b^{4} c^{2} d^{4} + 6 \, a b^{5} c d^{5} + b^{6} d^{6}\right )} \sqrt {x^{4} - x^{3}}}{x^{2}}} - {\left (a^{3} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + 3 \, a b^{2} c d^{4} + b^{3} d^{5}\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}}}{{\left (a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}\right )} x}\right ) - 3 \, d x^{3} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (c d^{5} - d^{6}\right )} x \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {3}{4}} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} + b^{3} d^{3}\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 3 \, d x^{3} \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (c d^{5} - d^{6}\right )} x \left (-\frac {a^{4} c^{4} + 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + b^{4} d^{4}}{c d^{7} - d^{8}}\right )^{\frac {3}{4}} - {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} + b^{3} d^{3}\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {3}{4}} a}{3 \, d x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 386, normalized size = 1.75 \begin {gather*} -\frac {\sqrt {2} {\left (a c + b d\right )} \log \left (-\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {c - d}{d}} + \sqrt {-\frac {1}{x} + 1}\right )}{2 \, {\left (c d^{3} - d^{4}\right )}^{\frac {1}{4}} d} - \frac {{\left (\sqrt {2} {\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} a c + \sqrt {2} {\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c - d}{d}\right )^{\frac {1}{4}}}\right )}{c d^{4} - d^{5}} - \frac {{\left (\sqrt {2} {\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} a c + \sqrt {2} {\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} b d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} - 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c - d}{d}\right )^{\frac {1}{4}}}\right )}{c d^{4} - d^{5}} + \frac {{\left ({\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} a c + {\left (c d^{3} - d^{4}\right )}^{\frac {3}{4}} b d\right )} \log \left (\sqrt {2} \left (\frac {c - d}{d}\right )^{\frac {1}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {\frac {c - d}{d}} + \sqrt {-\frac {1}{x} + 1}\right )}{\sqrt {2} c d^{4} - \sqrt {2} d^{5}} + \frac {4 \, a {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{4}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b x +a}{x \left (c x -d \right ) \left (x^{4}-x^{3}\right )^{\frac {1}{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 {b x + a}{{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (c x - d\right )} x}\,{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+b\,x}{x\,{\left (x^4-x^3\right )}^{1/4}\,\left (d-c\,x\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 + b x}{x \sqrt [4]{x^{3} \left (x - 1\right )} \left (c x - d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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