Optimal. Leaf size=206 \[ \frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{8 a^2 c^2 x}-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2} \]
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Rubi [A] time = 0.11, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {129, 151, 12, 93, 212, 208, 205} \begin {gather*} -\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}+\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{8 a^2 c^2 x}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 129
Rule 151
Rule 205
Rule 208
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}-\frac {\int \frac {\frac {1}{4} (7 b c+5 a d)+b d x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2 a c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}+\frac {\int \frac {21 b^2 c^2+6 a b c d+5 a^2 d^2}{16 x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2 a^2 c^2}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 a^2 c^2}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}+\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\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 )}{8 a^2 c^2}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \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 )}{16 a^{5/2} c^2}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \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 )}{16 a^{5/2} c^2}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 106, normalized size = 0.51 \begin {gather*} -\frac {\sqrt [4]{a+b x} \left (x^2 \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {c (a+b x)}{a (c+d x)}\right )+a (c+d x) (4 a c-5 a d x-7 b c x)\right )}{8 a^3 c^2 x^2 \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 269, normalized size = 1.31 \begin {gather*} \frac {\left (-5 a^2 d^2-6 a b c d-21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}+\frac {\left (-5 a^2 d^2-6 a b c d-21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}+\frac {\sqrt [4]{a+b x} \left (5 a^3 d^2-\frac {9 a^2 c d^2 (a+b x)}{c+d x}+6 a^2 b c d+\frac {7 b^2 c^3 (a+b x)}{c+d x}-11 a b^2 c^2+\frac {2 a b c^2 d (a+b x)}{c+d x}\right )}{8 a^2 c^2 \sqrt [4]{c+d x} \left (a-\frac {c (a+b x)}{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.83, size = 1528, normalized size = 7.42
result too large to display
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^{3}}\,{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^{3}}\, 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^{3}}\,{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 = -0.00 \begin {gather*} \int \frac {1}{x^3\,{\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^{3} \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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