Optimal. Leaf size=134 \[ \frac {(a d+3 b c) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac {(a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x} \]
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Rubi [A] time = 0.06, antiderivative size = 134, 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 = {96, 93, 212, 208, 205} \begin {gather*} \frac {(a d+3 b c) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac {(a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 205
Rule 208
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}-\frac {\left (\frac {3 b c}{4}+\frac {a d}{4}\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{a c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}-\frac {\left (4 \left (\frac {3 b c}{4}+\frac {a d}{4}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{a c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac {(3 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 a^{3/2} c}+\frac {(3 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 a^{3/2} c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac {(3 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^{7/4} c^{5/4}}+\frac {(3 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^{7/4} c^{5/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 72, normalized size = 0.54 \begin {gather*} \frac {\sqrt [4]{a+b x} \left (x (a d+3 b c) \, _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^2 c x \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 158, normalized size = 1.18 \begin {gather*} \frac {(a d+3 b c) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac {(a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac {\sqrt [4]{a+b x} (a d-b c)}{a c \sqrt [4]{c+d x} \left (a-\frac {c (a+b x)}{c+d x}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.59, size = 845, normalized size = 6.31 \begin {gather*} -\frac {4 \, a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (3 \, a^{5} b c^{5} + a^{6} c^{4} d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {3}{4}} - {\left (a^{5} c^{4} d x + a^{5} c^{5}\right )} \sqrt {\frac {{\left (9 \, b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} + {\left (a^{4} c^{2} d x + a^{4} c^{3}\right )} \sqrt {\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}}}{d x + c}} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {3}{4}}}{81 \, b^{4} c^{5} + 108 \, a b^{3} c^{4} d + 54 \, a^{2} b^{2} c^{3} d^{2} + 12 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4} + {\left (81 \, b^{4} c^{4} d + 108 \, a b^{3} c^{3} d^{2} + 54 \, a^{2} b^{2} c^{2} d^{3} + 12 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} x}\right ) - a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} 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 \, a c x} \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*} \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}} x^{2}}\, dx \end {gather*}
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*} \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{2}}\,{d x} \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 {1}{x^2\,{\left (a+b\,x\right )}^{3/4}\,{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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