Optimal. Leaf size=32 \[ -\frac {4 (c+d x)^{9/4}}{9 (b c-a d) (a+b x)^{9/4}} \]
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Rubi [A]
time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37}
\begin {gather*} -\frac {4 (c+d x)^{9/4}}{9 (a+b x)^{9/4} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx &=-\frac {4 (c+d x)^{9/4}}{9 (b c-a d) (a+b x)^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 32, normalized size = 1.00 \begin {gather*} -\frac {4 (c+d x)^{9/4}}{9 (b c-a d) (a+b x)^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 27, normalized size = 0.84
method | result | size |
gosper | \(\frac {4 \left (d x +c \right )^{\frac {9}{4}}}{9 \left (b x +a \right )^{\frac {9}{4}} \left (a d -b c \right )}\) | \(27\) |
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. 104 vs.
\(2 (26) = 52\).
time = 0.90, size = 104, normalized size = 3.25 \begin {gather*} -\frac {4 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{9 \, {\left (a^{3} b c - a^{4} d + {\left (b^{4} c - a b^{3} d\right )} x^{3} + 3 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2} + 3 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x\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 {\left (c + d x\right )^{\frac {5}{4}}}{\left (a + b x\right )^{\frac {13}{4}}}\, 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 [B]
time = 0.81, size = 99, normalized size = 3.09 \begin {gather*} \frac {4\,c^2\,{\left (c+d\,x\right )}^{1/4}+4\,d^2\,x^2\,{\left (c+d\,x\right )}^{1/4}+8\,c\,d\,x\,{\left (c+d\,x\right )}^{1/4}}{{\left (a+b\,x\right )}^{1/4}\,\left (9\,d\,a^3+18\,d\,a^2\,b\,x-9\,c\,a^2\,b+9\,d\,a\,b^2\,x^2-18\,c\,a\,b^2\,x-9\,c\,b^3\,x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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