Optimal. Leaf size=221 \[ -\frac {4 a \left (-x^3+x^4\right )^{3/4}}{3 d x^3}+\frac {\sqrt {2} (a c+b d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} x \sqrt [4]{-x^3+x^4}}{\sqrt {c-d} x^2-\sqrt {d} \sqrt {-x^3+x^4}}\right )}{\sqrt [4]{c-d} d^{7/4}}-\frac {\sqrt {2} (a c+b d) \tanh ^{-1}\left (\frac {\sqrt {c-d} x^2+\sqrt {d} \sqrt {-x^3+x^4}}{\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} x \sqrt [4]{-x^3+x^4}}\right )}{\sqrt [4]{c-d} d^{7/4}} \]
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Rubi [A]
time = 0.24, 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 = {2081, 160, 12,
95, 218, 214, 211} \begin {gather*} -\frac {2 \sqrt [4]{x-1} x^{3/4} (a c+b d) \text {ArcTan}\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 95
Rule 160
Rule 211
Rule 214
Rule 218
Rule 2081
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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 5.60, size = 229, normalized size = 1.04 \begin {gather*} \frac {-4 a \sqrt [4]{c-d} d^{3/4} (-1+x)+3 \sqrt {2} (a c+b d) \sqrt [4]{-1+x} x^{3/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} \sqrt [4]{-1+x} \sqrt [4]{x}}{-\sqrt {d} \sqrt {-1+x}+\sqrt {c-d} \sqrt {x}}\right )-3 \sqrt {2} (a c+b d) \sqrt [4]{-1+x} x^{3/4} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {-1+x}+\sqrt {c-d} \sqrt {x}}{\sqrt {2} \sqrt [4]{c-d} \sqrt [4]{d} \sqrt [4]{-1+x} \sqrt [4]{x}}\right )}{3 \sqrt [4]{c-d} d^{7/4} \sqrt [4]{(-1+x) x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {b x +a}{x \left (c x -d \right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1021 vs.
\(2 (178) = 356\).
time = 0.50, 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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 386 vs.
\(2 (178) = 356\).
time = 0.41, 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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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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