Optimal. Leaf size=104 \[ -\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}+\frac {2 \sqrt [4]{b c-a d} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{5/4} \sqrt {a+b x}} \]
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Rubi [A]
time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {49, 65, 230,
227} \begin {gather*} \frac {2 \sqrt [4]{b c-a d} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{5/4} \sqrt {a+b x}}-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 227
Rule 230
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{c+d x}}{(a+b x)^{3/2}} \, dx &=-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}+\frac {d \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{2 b}\\ &=-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b}\\ &=-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}+\frac {\left (2 \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b \sqrt {a+b x}}\\ &=-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}+\frac {2 \sqrt [4]{b c-a d} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{5/4} \sqrt {a+b x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 71, normalized size = 0.68 \begin {gather*} -\frac {2 \sqrt [4]{c+d x} \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {1}{2};\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} \sqrt [4]{\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {3}{2}}}\, dx\]
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 {\sqrt [4]{c + d x}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 )}^{1/4}}{{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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