Optimal. Leaf size=167 \[ -\frac {5 (b c-a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d}+\frac {5 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}}+\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}} \]
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Rubi [A]
time = 0.07, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {52, 65, 246,
218, 214, 211} \begin {gather*} \frac {5 (b c-a d)^2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}}+\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}}-\frac {5 \sqrt [4]{a+b x} (c+d x)^{3/4} (b c-a d)}{8 d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 211
Rule 214
Rule 218
Rule 246
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/4}}{\sqrt [4]{c+d x}} \, dx &=\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d}-\frac {(5 (b c-a d)) \int \frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx}{8 d}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 d^2}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d}+\frac {\left (5 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{8 b d^2}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d}+\frac {\left (5 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{8 b d^2}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d}+\frac {\left (5 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 \sqrt {b} d^2}+\frac {\left (5 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 \sqrt {b} d^2}\\ &=-\frac {5 (b c-a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 d}+\frac {5 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}}+\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{3/4} d^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 143, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt [4]{d} \sqrt [4]{a+b x} (c+d x)^{3/4} (-5 b c+9 a d+4 b d x)+\frac {5 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4}}+\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4}}}{16 d^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {5}{4}}}{\left (d x +c \right )^{\frac {1}{4}}}\, dx\]
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. 1468 vs.
\(2 (127) = 254\).
time = 0.78, size = 1468, normalized size = 8.79 \begin {gather*} -\frac {20 \, d^{2} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b^{4} c^{2} d^{7} - 2 \, a b^{3} c d^{8} + a^{2} b^{2} d^{9}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}\right )^{\frac {3}{4}} - {\left (b^{2} d^{8} x + b^{2} c d^{7}\right )} \sqrt {\frac {{\left (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}\right )} \sqrt {b x + a} \sqrt {d x + c} + {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )} \sqrt {\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}}}{d x + c}} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}\right )^{\frac {3}{4}}}{b^{8} c^{9} - 8 \, a b^{7} c^{8} d + 28 \, a^{2} b^{6} c^{7} d^{2} - 56 \, a^{3} b^{5} c^{6} d^{3} + 70 \, a^{4} b^{4} c^{5} d^{4} - 56 \, a^{5} b^{3} c^{4} d^{5} + 28 \, a^{6} b^{2} c^{3} d^{6} - 8 \, a^{7} b c^{2} d^{7} + a^{8} c d^{8} + {\left (b^{8} c^{8} d - 8 \, a b^{7} c^{7} d^{2} + 28 \, a^{2} b^{6} c^{6} d^{3} - 56 \, a^{3} b^{5} c^{5} d^{4} + 70 \, a^{4} b^{4} c^{4} d^{5} - 56 \, a^{5} b^{3} c^{3} d^{6} + 28 \, a^{6} b^{2} c^{2} d^{7} - 8 \, a^{7} b c d^{8} + a^{8} d^{9}\right )} x}\right ) - 5 \, d^{2} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}\right )^{\frac {1}{4}} \log \left (\frac {5 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (b d^{3} x + b c d^{2}\right )} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) + 5 \, d^{2} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}\right )^{\frac {1}{4}} \log \left (\frac {5 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (b d^{3} x + b c d^{2}\right )} \left (\frac {b^{8} c^{8} - 8 \, a b^{7} c^{7} d + 28 \, a^{2} b^{6} c^{6} d^{2} - 56 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} - 56 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} - 8 \, a^{7} b c d^{7} + a^{8} d^{8}}{b^{3} d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) - 4 \, {\left (4 \, b d x - 5 \, b c + 9 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{32 \, d^{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 {\left (a + b x\right )^{\frac {5}{4}}}{\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 )}^{5/4}}{{\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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