Optimal. Leaf size=705 \[ \frac {2 (a+b x)^{3/4} (c+d x)^{3/4}}{3 d}-\frac {\sqrt {(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{\sqrt {b} d^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )}+\frac {(b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{\sqrt {2} b^{3/4} d^{7/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {(b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} b^{3/4} d^{7/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}} \]
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Rubi [A]
time = 0.44, antiderivative size = 705, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {52, 64, 637,
311, 226, 1210} \begin {gather*} -\frac {(b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} b^{3/4} d^{7/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}+\frac {(b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{\sqrt {2} b^{3/4} d^{7/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {\sqrt {(a+b x) (c+d x)} \sqrt {(a d+b c+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{\sqrt {b} d^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (a d+b c+2 b d x) \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )}+\frac {2 (a+b x)^{3/4} (c+d x)^{3/4}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 64
Rule 226
Rule 311
Rule 637
Rule 1210
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/4}}{\sqrt [4]{c+d x}} \, dx &=\frac {2 (a+b x)^{3/4} (c+d x)^{3/4}}{3 d}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx}{2 d}\\ &=\frac {2 (a+b x)^{3/4} (c+d x)^{3/4}}{3 d}-\frac {\left ((b c-a d) \sqrt [4]{(a+b x) (c+d x)}\right ) \int \frac {1}{\sqrt [4]{a c+(b c+a d) x+b d x^2}} \, dx}{2 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}\\ &=\frac {2 (a+b x)^{3/4} (c+d x)^{3/4}}{3 d}-\frac {\left (2 (b c-a d) \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{d \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)}\\ &=\frac {2 (a+b x)^{3/4} (c+d x)^{3/4}}{3 d}-\frac {\left ((b c-a d)^2 \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{\sqrt {b} d^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)}+\frac {\left ((b c-a d)^2 \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {1-\frac {2 \sqrt {b} \sqrt {d} x^2}{b c-a d}}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{\sqrt {b} d^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)}\\ &=\frac {2 (a+b x)^{3/4} (c+d x)^{3/4}}{3 d}-\frac {\sqrt {(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{\sqrt {b} d^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )}+\frac {(b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{\sqrt {2} b^{3/4} d^{7/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {(b c-a d)^{5/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} b^{3/4} d^{7/4} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 73, normalized size = 0.10 \begin {gather*} \frac {4 (a+b x)^{7/4} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{4},\frac {7}{4};\frac {11}{4};\frac {d (a+b x)}{-b c+a d}\right )}{7 b \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {3}{4}}}{\left (d x +c \right )^{\frac {1}{4}}}\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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 = -0.00 \begin {gather*} \int \frac {{\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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