Optimal. Leaf size=194 \[ \frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}+\frac {(b c-a d) (3 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {(b c-a d) (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}} \]
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Rubi [A]
time = 0.06, 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 = {98, 96, 95, 218,
214, 211} \begin {gather*} \frac {(b c-a d) (5 a d+3 b c) \text {ArcTan}\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 95
Rule 96
Rule 98
Rule 211
Rule 214
Rule 218
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)) \text {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)) \text {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)) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.05, size = 106, normalized size = 0.55 \begin {gather*} \frac {\sqrt [4]{a+b x} \left (-a (c+d x) (4 a c+b c x-5 a d x)+\left (3 b^2 c^2+2 a b c d-5 a^2 d^2\right ) x^2 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {c (a+b x)}{a (c+d x)}\right )\right )}{8 a^2 c^2 x^2 \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{3} \left (d x +c \right )^{\frac {1}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1520 vs.
\(2 (154) = 308\).
time = 0.56, size = 1520, normalized size = 7.84 \begin {gather*} \frac {4 \, a c^{2} x^{2} \left (\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (3 \, a^{5} b^{2} c^{9} + 2 \, a^{6} b c^{8} d - 5 \, a^{7} c^{7} d^{2}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} \left (\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}\right )^{\frac {3}{4}} + {\left (a^{5} c^{7} d x + a^{5} c^{8}\right )} \sqrt {\frac {{\left (9 \, b^{4} c^{4} + 12 \, a b^{3} c^{3} d - 26 \, a^{2} b^{2} c^{2} d^{2} - 20 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4}\right )} \sqrt {b x + a} \sqrt {d x + c} + {\left (a^{4} c^{4} d x + a^{4} c^{5}\right )} \sqrt {\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}}}{d x + c}} \left (\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}\right )^{\frac {3}{4}}}{81 \, b^{8} c^{9} + 216 \, a b^{7} c^{8} d - 324 \, a^{2} b^{6} c^{7} d^{2} - 984 \, a^{3} b^{5} c^{6} d^{3} + 646 \, a^{4} b^{4} c^{5} d^{4} + 1640 \, a^{5} b^{3} c^{4} d^{5} - 900 \, a^{6} b^{2} c^{3} d^{6} - 1000 \, a^{7} b c^{2} d^{7} + 625 \, a^{8} c d^{8} + {\left (81 \, b^{8} c^{8} d + 216 \, a b^{7} c^{7} d^{2} - 324 \, a^{2} b^{6} c^{6} d^{3} - 984 \, a^{3} b^{5} c^{5} d^{4} + 646 \, a^{4} b^{4} c^{4} d^{5} + 1640 \, a^{5} b^{3} c^{3} d^{6} - 900 \, a^{6} b^{2} c^{2} d^{7} - 1000 \, a^{7} b c d^{8} + 625 \, a^{8} d^{9}\right )} x}\right ) + a c^{2} x^{2} \left (\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d - 5 \, a^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (a^{2} c^{2} d x + a^{2} c^{3}\right )} \left (\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}\right )^{\frac {1}{4}}}{d x + c}\right ) - a c^{2} x^{2} \left (\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d - 5 \, a^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (a^{2} c^{2} d x + a^{2} c^{3}\right )} \left (\frac {81 \, b^{8} c^{8} + 216 \, a b^{7} c^{7} d - 324 \, a^{2} b^{6} c^{6} d^{2} - 984 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} + 1640 \, a^{5} b^{3} c^{3} d^{5} - 900 \, a^{6} b^{2} c^{2} d^{6} - 1000 \, a^{7} b c d^{7} + 625 \, a^{8} d^{8}}{a^{7} c^{9}}\right )^{\frac {1}{4}}}{d x + c}\right ) - 4 \, {\left (4 \, a c + {\left (b c - 5 \, a d\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{32 \, a c^{2} x^{2}} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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