Optimal. Leaf size=194 \[ \frac {(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {(b c-a d) (5 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+3 b c)}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {96, 94, 93, 212, 208, 205} \begin {gather*} \frac {(b c-a d) (5 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 )}{16 a^{7/4} c^{9/4}}+\frac {(b c-a d) (5 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 )}{16 a^{7/4} c^{9/4}}+\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+3 b c)}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 205
Rule 208
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx &=-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}-\frac {\left (\frac {3 b c}{4}+\frac {5 a d}{4}\right ) \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx}{2 a c}\\ &=\frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}-\frac {((b c-a d) (3 b c+5 a d)) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 a c^2}\\ &=\frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}-\frac {((b c-a d) (3 b c+5 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 )}{8 a c^2}\\ &=\frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}+\frac {((b c-a d) (3 b c+5 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 )}{16 a^{3/2} c^2}+\frac {((b c-a d) (3 b c+5 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 )}{16 a^{3/2} c^2}\\ &=\frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}+\frac {(b c-a d) (3 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {(b c-a d) (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 106, normalized size = 0.55 \begin {gather*} \frac {\sqrt [4]{a+b x} \left (x^2 \left (-5 a^2 d^2+2 a b c d+3 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+b c x)\right )}{8 a^2 c^2 x^2 \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 116.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.22, size = 1520, normalized size = 7.84
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 {{\left (b x + a\right )}^{\frac {1}{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 = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x +a \right )^{\frac {1}{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 {{\left (b x + a\right )}^{\frac {1}{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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{1/4}}{x^3\,{\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 {\sqrt [4]{a + b x}}{x^{3} \sqrt [4]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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