Optimal. Leaf size=101 \[ -\frac{128 d^2 (c+d x)^{9/4}}{1989 (a+b x)^{9/4} (b c-a d)^3}+\frac{32 d (c+d x)^{9/4}}{221 (a+b x)^{13/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)} \]
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Rubi [A] time = 0.0173348, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{128 d^2 (c+d x)^{9/4}}{1989 (a+b x)^{9/4} (b c-a d)^3}+\frac{32 d (c+d x)^{9/4}}{221 (a+b x)^{13/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx &=-\frac{4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}-\frac{(8 d) \int \frac{(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx}{17 (b c-a d)}\\ &=-\frac{4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}+\frac{32 d (c+d x)^{9/4}}{221 (b c-a d)^2 (a+b x)^{13/4}}+\frac{\left (32 d^2\right ) \int \frac{(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx}{221 (b c-a d)^2}\\ &=-\frac{4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}+\frac{32 d (c+d x)^{9/4}}{221 (b c-a d)^2 (a+b x)^{13/4}}-\frac{128 d^2 (c+d x)^{9/4}}{1989 (b c-a d)^3 (a+b x)^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.0468354, size = 77, normalized size = 0.76 \[ -\frac{4 (c+d x)^{9/4} \left (221 a^2 d^2+34 a b d (4 d x-9 c)+b^2 \left (117 c^2-72 c d x+32 d^2 x^2\right )\right )}{1989 (a+b x)^{17/4} (b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 105, normalized size = 1. \begin{align*}{\frac{128\,{b}^{2}{d}^{2}{x}^{2}+544\,ab{d}^{2}x-288\,{b}^{2}cdx+884\,{a}^{2}{d}^{2}-1224\,abcd+468\,{b}^{2}{c}^{2}}{1989\,{a}^{3}{d}^{3}-5967\,{a}^{2}cb{d}^{2}+5967\,a{b}^{2}{c}^{2}d-1989\,{b}^{3}{c}^{3}} \left ( dx+c \right ) ^{{\frac{9}{4}}} \left ( bx+a \right ) ^{-{\frac{17}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{21}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.57569, size = 879, normalized size = 8.7 \begin{align*} -\frac{4 \,{\left (32 \, b^{2} d^{4} x^{4} + 117 \, b^{2} c^{4} - 306 \, a b c^{3} d + 221 \, a^{2} c^{2} d^{2} - 8 \,{\left (b^{2} c d^{3} - 17 \, a b d^{4}\right )} x^{3} +{\left (5 \, b^{2} c^{2} d^{2} - 34 \, a b c d^{3} + 221 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (81 \, b^{2} c^{3} d - 238 \, a b c^{2} d^{2} + 221 \, a^{2} c d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{1989 \,{\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} +{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \,{\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{21}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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