Optimal. Leaf size=167 \[ \frac {5 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 b^2}+\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b}-\frac {5 (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{9/4} d^{3/4}}+\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{9/4} d^{3/4}} \]
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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, 338,
304, 211, 214} \begin {gather*} -\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^{9/4} d^{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 )}{16 b^{9/4} d^{3/4}}+\frac {5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)}{8 b^2}+\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 211
Rule 214
Rule 304
Rule 338
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/4}}{\sqrt [4]{a+b x}} \, dx &=\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b}+\frac {(5 (b c-a d)) \int \frac {\sqrt [4]{c+d x}}{\sqrt [4]{a+b x}} \, dx}{8 b}\\ &=\frac {5 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 b^2}+\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b}+\frac {\left (5 (b c-a d)^2\right ) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{32 b^2}\\ &=\frac {5 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 b^2}+\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b}+\frac {\left (5 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (c-\frac {a d}{b}+\frac {d x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{a+b x}\right )}{8 b^3}\\ &=\frac {5 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 b^2}+\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b}+\frac {\left (5 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {x^2}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{8 b^3}\\ &=\frac {5 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 b^2}+\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b}+\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 b^2 \sqrt {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 b^2 \sqrt {d}}\\ &=\frac {5 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 b^2}+\frac {(a+b x)^{3/4} (c+d x)^{5/4}}{2 b}-\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^{9/4} d^{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 )}{16 b^{9/4} d^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 143, normalized size = 0.86 \begin {gather*} \frac {(b c-a d)^2 \left (\frac {2 \sqrt [4]{b} (a+b x)^{3/4} \sqrt [4]{c+d x} (9 b c-5 a d+4 b d x)}{(b c-a d)^2}+\frac {5 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{d^{3/4}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{d^{3/4}}\right )}{16 b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\left (b x +a \right )^{\frac {1}{4}}}\, dx\]
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 = 0.38, size = 1468, normalized size = 8.79
result too large to display
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 (c + d x\right )^{\frac {5}{4}}}{\sqrt [4]{a + b x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 (c+d\,x\right )}^{5/4}}{{\left (a+b\,x\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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