Optimal. Leaf size=85 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \]
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Rubi [A] time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {63, 331, 298, 205, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 208
Rule 298
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx &=\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{\left (c-\frac {a d}{b}+\frac {d x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{a+b x}\right )}{b}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{b}\\ &=\frac {2 \operatorname {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 )}{\sqrt {d}}-\frac {2 \operatorname {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 )}{\sqrt {d}}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.86 \[ \frac {4 (a+b x)^{3/4} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {d (a+b x)}{a d-b c}\right )}{3 b (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 234, normalized size = 2.75 \[ -4 \, \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} b d^{2} \left (\frac {1}{b d^{3}}\right )^{\frac {3}{4}} - {\left (b^{2} d^{2} x + a b d^{2}\right )} \sqrt {\frac {{\left (b d^{2} x + a d^{2}\right )} \sqrt {\frac {1}{b d^{3}}} + \sqrt {b x + a} \sqrt {d x + c}}{b x + a}} \left (\frac {1}{b d^{3}}\right )^{\frac {3}{4}}}{b x + a}\right ) + \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) - \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {1}{4}} \left (d x +c \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,x\right )}^{1/4}\,{\left (c+d\,x\right )}^{3/4}} \,d x \]
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 \[ \int \frac {1}{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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