Optimal. Leaf size=169 \[ -\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}} \]
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Rubi [A]
time = 0.07, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {132, 65, 246,
218, 214, 211, 12, 95} \begin {gather*} -\frac {2 \sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 95
Rule 132
Rule 211
Rule 214
Rule 218
Rule 246
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx &=a \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx+b \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\\ &=4 \text {Subst}\left (\int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )+(4 a) \text {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )\\ &=4 \text {Subst}\left (\int \frac {1}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )-\left (2 \sqrt {a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {c} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )-\left (2 \sqrt {a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+\sqrt {c} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )\\ &=-\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}-\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\left (2 \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )+\left (2 \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )\\ &=-\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.05, size = 97, normalized size = 0.57 \begin {gather*} \frac {4 \sqrt [4]{a+b x} \left (\sqrt [4]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {d (a+b x)}{-b c+a d}\right )-\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {c (a+b x)}{a (c+d x)}\right )\right )}{\sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x \left (d x +c \right )^{\frac {1}{4}}}\, dx\]
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. 396 vs.
\(2 (121) = 242\).
time = 0.61, size = 396, normalized size = 2.34 \begin {gather*} 4 \, \left (\frac {a}{c}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} c \left (\frac {a}{c}\right )^{\frac {3}{4}} - {\left (c d x + c^{2}\right )} \sqrt {\frac {{\left (d x + c\right )} \sqrt {\frac {a}{c}} + \sqrt {b x + a} \sqrt {d x + c}}{d x + c}} \left (\frac {a}{c}\right )^{\frac {3}{4}}}{a d x + a c}\right ) - 4 \, \left (\frac {b}{d}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} d \left (\frac {b}{d}\right )^{\frac {3}{4}} - {\left (d^{2} x + c d\right )} \sqrt {\frac {{\left (d x + c\right )} \sqrt {\frac {b}{d}} + \sqrt {b x + a} \sqrt {d x + c}}{d x + c}} \left (\frac {b}{d}\right )^{\frac {3}{4}}}{b d x + b c}\right ) - \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) \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]{a + b x}}{x \sqrt [4]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (a+b\,x\right )}^{1/4}}{x\,{\left (c+d\,x\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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