Optimal. Leaf size=136 \[ \frac{512 d^3 (c+d x)^{9/4}}{13923 (a+b x)^{9/4} (b c-a d)^4}-\frac{128 d^2 (c+d x)^{9/4}}{1547 (a+b x)^{13/4} (b c-a d)^3}+\frac{16 d (c+d x)^{9/4}}{119 (a+b x)^{17/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)} \]
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Rubi [A] time = 0.0289592, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{512 d^3 (c+d x)^{9/4}}{13923 (a+b x)^{9/4} (b c-a d)^4}-\frac{128 d^2 (c+d x)^{9/4}}{1547 (a+b x)^{13/4} (b c-a d)^3}+\frac{16 d (c+d x)^{9/4}}{119 (a+b x)^{17/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)} \]
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)^{25/4}} \, dx &=-\frac{4 (c+d x)^{9/4}}{21 (b c-a d) (a+b x)^{21/4}}-\frac{(4 d) \int \frac{(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx}{7 (b c-a d)}\\ &=-\frac{4 (c+d x)^{9/4}}{21 (b c-a d) (a+b x)^{21/4}}+\frac{16 d (c+d x)^{9/4}}{119 (b c-a d)^2 (a+b x)^{17/4}}+\frac{\left (32 d^2\right ) \int \frac{(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx}{119 (b c-a d)^2}\\ &=-\frac{4 (c+d x)^{9/4}}{21 (b c-a d) (a+b x)^{21/4}}+\frac{16 d (c+d x)^{9/4}}{119 (b c-a d)^2 (a+b x)^{17/4}}-\frac{128 d^2 (c+d x)^{9/4}}{1547 (b c-a d)^3 (a+b x)^{13/4}}-\frac{\left (128 d^3\right ) \int \frac{(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx}{1547 (b c-a d)^3}\\ &=-\frac{4 (c+d x)^{9/4}}{21 (b c-a d) (a+b x)^{21/4}}+\frac{16 d (c+d x)^{9/4}}{119 (b c-a d)^2 (a+b x)^{17/4}}-\frac{128 d^2 (c+d x)^{9/4}}{1547 (b c-a d)^3 (a+b x)^{13/4}}+\frac{512 d^3 (c+d x)^{9/4}}{13923 (b c-a d)^4 (a+b x)^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.0638237, size = 118, normalized size = 0.87 \[ \frac{4 (c+d x)^{9/4} \left (357 a^2 b d^2 (4 d x-9 c)+1547 a^3 d^3+21 a b^2 d \left (117 c^2-72 c d x+32 d^2 x^2\right )+b^3 \left (468 c^2 d x-663 c^3-288 c d^2 x^2+128 d^3 x^3\right )\right )}{13923 (a+b x)^{21/4} (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 171, normalized size = 1.3 \begin{align*}{\frac{512\,{b}^{3}{d}^{3}{x}^{3}+2688\,a{b}^{2}{d}^{3}{x}^{2}-1152\,{b}^{3}c{d}^{2}{x}^{2}+5712\,{a}^{2}b{d}^{3}x-6048\,a{b}^{2}c{d}^{2}x+1872\,{b}^{3}{c}^{2}dx+6188\,{a}^{3}{d}^{3}-12852\,{a}^{2}cb{d}^{2}+9828\,a{b}^{2}{c}^{2}d-2652\,{b}^{3}{c}^{3}}{13923\,{a}^{4}{d}^{4}-55692\,{a}^{3}bc{d}^{3}+83538\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-55692\,a{b}^{3}{c}^{3}d+13923\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{9}{4}}} \left ( bx+a \right ) ^{-{\frac{21}{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{25}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.62618, size = 1354, normalized size = 9.96 \begin{align*} \frac{4 \,{\left (128 \, b^{3} d^{5} x^{5} - 663 \, b^{3} c^{5} + 2457 \, a b^{2} c^{4} d - 3213 \, a^{2} b c^{3} d^{2} + 1547 \, a^{3} c^{2} d^{3} - 32 \,{\left (b^{3} c d^{4} - 21 \, a b^{2} d^{5}\right )} x^{4} + 4 \,{\left (5 \, b^{3} c^{2} d^{3} - 42 \, a b^{2} c d^{4} + 357 \, a^{2} b d^{5}\right )} x^{3} -{\left (15 \, b^{3} c^{3} d^{2} - 105 \, a b^{2} c^{2} d^{3} + 357 \, a^{2} b c d^{4} - 1547 \, a^{3} d^{5}\right )} x^{2} - 2 \,{\left (429 \, b^{3} c^{4} d - 1701 \, a b^{2} c^{3} d^{2} + 2499 \, a^{2} b c^{2} d^{3} - 1547 \, a^{3} c d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{13923 \,{\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} +{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \,{\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \,{\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \,{\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \,{\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \,{\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\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{25}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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