Optimal. Leaf size=85 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4} \sqrt [4]{d}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4} \sqrt [4]{d}} \]
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Rubi [A]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {65, 246, 218,
214, 211} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{b^{3/4} \sqrt [4]{d}}+\frac {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} \sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 211
Rule 214
Rule 218
Rule 246
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=\frac {4 \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 )}{b}\\ &=\frac {4 \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 )}{b}\\ &=\frac {2 \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 )}{\sqrt {b}}+\frac {2 \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 )}{\sqrt {b}}\\ &=\frac {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} \sqrt [4]{d}}+\frac {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} \sqrt [4]{d}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 73, normalized size = 0.86 \begin {gather*} \frac {2 \left (\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )\right )}{b^{3/4} \sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}\, dx\]
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. 234 vs.
\(2 (61) = 122\).
time = 0.63, size = 234, normalized size = 2.75 \begin {gather*} -4 \, \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} b^{2} d \left (\frac {1}{b^{3} d}\right )^{\frac {3}{4}} - {\left (b^{2} d^{2} x + b^{2} c d\right )} \sqrt {\frac {{\left (b^{2} d x + b^{2} c\right )} \sqrt {\frac {1}{b^{3} d}} + \sqrt {b x + a} \sqrt {d x + c}}{d x + c}} \left (\frac {1}{b^{3} d}\right )^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{3} d}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) \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}{\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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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 {1}{{\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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