Optimal. Leaf size=66 \[ \frac {16 d (c+d x)^{9/4}}{117 (a+b x)^{9/4} (b c-a d)^2}-\frac {4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {16 d (c+d x)^{9/4}}{117 (a+b x)^{9/4} (b c-a d)^2}-\frac {4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx &=-\frac {4 (c+d x)^{9/4}}{13 (b c-a d) (a+b x)^{13/4}}-\frac {(4 d) \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx}{13 (b c-a d)}\\ &=-\frac {4 (c+d x)^{9/4}}{13 (b c-a d) (a+b x)^{13/4}}+\frac {16 d (c+d x)^{9/4}}{117 (b c-a d)^2 (a+b x)^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {4 (c+d x)^{9/4} (13 a d-9 b c+4 b d x)}{117 (a+b x)^{13/4} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 51, normalized size = 0.77 \begin {gather*} -\frac {4 (c+d x)^{9/4} \left (\frac {9 b (c+d x)}{a+b x}-13 d\right )}{117 (a+b x)^{9/4} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 235, normalized size = 3.56 \begin {gather*} \frac {4 \, {\left (4 \, b d^{3} x^{3} - 9 \, b c^{3} + 13 \, a c^{2} d - {\left (b c d^{2} - 13 \, a d^{3}\right )} x^{2} - 2 \, {\left (7 \, b c^{2} d - 13 \, a c d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{117 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 4 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{2} + 4 \, {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x\right )}} \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*} \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {17}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.82 \begin {gather*} \frac {4 \left (d x +c \right )^{\frac {9}{4}} \left (4 b d x +13 a d -9 b c \right )}{117 \left (b x +a \right )^{\frac {13}{4}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \end {gather*}
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*} \int \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {17}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 178, normalized size = 2.70 \begin {gather*} \frac {{\left (c+d\,x\right )}^{1/4}\,\left (\frac {16\,d^3\,x^3}{117\,b^2\,{\left (a\,d-b\,c\right )}^2}-\frac {36\,b\,c^3-52\,a\,c^2\,d}{117\,b^3\,{\left (a\,d-b\,c\right )}^2}+\frac {x^2\,\left (52\,a\,d^3-4\,b\,c\,d^2\right )}{117\,b^3\,{\left (a\,d-b\,c\right )}^2}+\frac {8\,c\,d\,x\,\left (13\,a\,d-7\,b\,c\right )}{117\,b^3\,{\left (a\,d-b\,c\right )}^2}\right )}{x^3\,{\left (a+b\,x\right )}^{1/4}+\frac {a^3\,{\left (a+b\,x\right )}^{1/4}}{b^3}+\frac {3\,a\,x^2\,{\left (a+b\,x\right )}^{1/4}}{b}+\frac {3\,a^2\,x\,{\left (a+b\,x\right )}^{1/4}}{b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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