Optimal. Leaf size=85 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}} \]
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Rubi [A] time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {93, 212, 208, 205} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 205
Rule 208
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=4 \operatorname {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )\\ &=-\frac {2 \operatorname {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 )}{\sqrt {a}}-\frac {2 \operatorname {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 )}{\sqrt {a}}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 48, normalized size = 0.56 \[ -\frac {4 \sqrt [4]{a+b x} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {c (a+b x)}{a (c+d x)}\right )}{a \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 234, normalized size = 2.75 \[ 4 \, \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} a^{2} c \left (\frac {1}{a^{3} c}\right )^{\frac {3}{4}} - {\left (a^{2} c d x + a^{2} c^{2}\right )} \sqrt {\frac {{\left (a^{2} d x + a^{2} c\right )} \sqrt {\frac {1}{a^{3} c}} + \sqrt {b x + a} \sqrt {d x + c}}{d x + c}} \left (\frac {1}{a^{3} c}\right )^{\frac {3}{4}}}{d x + c}\right ) - \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a d x + a c\right )} \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a d x + a c\right )} \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}} x}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]
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 \[ \int \frac {1}{x \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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