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 = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \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)^{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*}
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Mathematica [A] time = 0.07, size = 118, normalized size = 0.87 \begin {gather*} \frac {4 (c+d x)^{9/4} \left (1547 a^3 d^3+357 a^2 b d^2 (4 d x-9 c)+21 a b^2 d \left (117 c^2-72 c d x+32 d^2 x^2\right )+b^3 \left (-663 c^3+468 c^2 d x-288 c d^2 x^2+128 d^3 x^3\right )\right )}{13923 (a+b x)^{21/4} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 95, normalized size = 0.70 \begin {gather*} -\frac {4 (c+d x)^{9/4} \left (\frac {663 b^3 (c+d x)^3}{(a+b x)^3}-\frac {2457 b^2 d (c+d x)^2}{(a+b x)^2}+\frac {3213 b d^2 (c+d x)}{a+b x}-1547 d^3\right )}{13923 (a+b x)^{9/4} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.66, size = 649, normalized size = 4.77 \begin {gather*} \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 {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 {25}{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 = 171, normalized size = 1.26 \begin {gather*} \frac {4 \left (d x +c \right )^{\frac {9}{4}} \left (128 b^{3} d^{3} x^{3}+672 a \,b^{2} d^{3} x^{2}-288 b^{3} c \,d^{2} x^{2}+1428 a^{2} b \,d^{3} x -1512 a \,b^{2} c \,d^{2} x +468 b^{3} c^{2} d x +1547 a^{3} d^{3}-3213 a^{2} b c \,d^{2}+2457 a \,b^{2} c^{2} d -663 b^{3} c^{3}\right )}{13923 \left (b x +a \right )^{\frac {21}{4}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\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 {25}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 376, normalized size = 2.76 \begin {gather*} \frac {{\left (c+d\,x\right )}^{1/4}\,\left (\frac {x^2\,\left (6188\,a^3\,d^5-1428\,a^2\,b\,c\,d^4+420\,a\,b^2\,c^2\,d^3-60\,b^3\,c^3\,d^2\right )}{13923\,b^5\,{\left (a\,d-b\,c\right )}^4}-\frac {-6188\,a^3\,c^2\,d^3+12852\,a^2\,b\,c^3\,d^2-9828\,a\,b^2\,c^4\,d+2652\,b^3\,c^5}{13923\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (12376\,a^3\,c\,d^4-19992\,a^2\,b\,c^2\,d^3+13608\,a\,b^2\,c^3\,d^2-3432\,b^3\,c^4\,d\right )}{13923\,b^5\,{\left (a\,d-b\,c\right )}^4}+\frac {512\,d^5\,x^5}{13923\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {128\,d^4\,x^4\,\left (21\,a\,d-b\,c\right )}{13923\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^3\,x^3\,\left (357\,a^2\,d^2-42\,a\,b\,c\,d+5\,b^2\,c^2\right )}{13923\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5\,{\left (a+b\,x\right )}^{1/4}+\frac {a^5\,{\left (a+b\,x\right )}^{1/4}}{b^5}+\frac {10\,a^2\,x^3\,{\left (a+b\,x\right )}^{1/4}}{b^2}+\frac {10\,a^3\,x^2\,{\left (a+b\,x\right )}^{1/4}}{b^3}+\frac {5\,a\,x^4\,{\left (a+b\,x\right )}^{1/4}}{b}+\frac {5\,a^4\,x\,{\left (a+b\,x\right )}^{1/4}}{b^4}} \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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