Optimal. Leaf size=131 \[ -\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x} \]
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Rubi [A] time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {94, 93, 212, 208, 205} \[ -\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 205
Rule 208
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}+\frac {(b c-a d) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac {(b c-a d) \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 )}{2 \sqrt {a} c}-\frac {(b c-a d) \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 )}{2 \sqrt {a} c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 73, normalized size = 0.56 \[ \frac {\sqrt [4]{a+b x} \left ((a d x-b c x) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {c (a+b x)}{a (c+d x)}\right )-a (c+d x)\right )}{a c x \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 820, normalized size = 6.26 \[ -\frac {4 \, c x \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {3}{4}} + {\left (a^{2} c^{4} d x + a^{2} c^{5}\right )} \sqrt {\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} + {\left (a^{2} c^{2} d x + a^{2} c^{3}\right )} \sqrt {\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}}}{d x + c}} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {3}{4}}}{b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4} + {\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} x}\right ) + c x \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (a c d x + a c^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - c x \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (a c d x + a c^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + 4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{4 \, c x} \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}} x^{2}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{1/4}}{x^2\,{\left (c+d\,x\right )}^{1/4}} \,d x \]
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 \[ \int \frac {\sqrt [4]{a + b x}}{x^{2} \sqrt [4]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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