Optimal. Leaf size=158 \[ -\frac {10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac {20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac {20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac {10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac {2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \]
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Rubi [A] time = 0.05, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \begin {gather*} -\frac {10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac {20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac {20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac {10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac {2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^5 (c+d x)^{3/2} \, dx &=\int \left (\frac {(-b c+a d)^5 (c+d x)^{3/2}}{d^5}+\frac {5 b (b c-a d)^4 (c+d x)^{5/2}}{d^5}-\frac {10 b^2 (b c-a d)^3 (c+d x)^{7/2}}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{9/2}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{11/2}}{d^5}+\frac {b^5 (c+d x)^{13/2}}{d^5}\right ) \, dx\\ &=-\frac {2 (b c-a d)^5 (c+d x)^{5/2}}{5 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{7/2}}{7 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{9/2}}{9 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{11/2}}{11 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{13/2}}{13 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 123, normalized size = 0.78 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (-17325 b^4 (c+d x)^4 (b c-a d)+40950 b^3 (c+d x)^3 (b c-a d)^2-50050 b^2 (c+d x)^2 (b c-a d)^3+32175 b (c+d x) (b c-a d)^4-9009 (b c-a d)^5+3003 b^5 (c+d x)^5\right )}{45045 d^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 315, normalized size = 1.99 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (9009 a^5 d^5+32175 a^4 b d^4 (c+d x)-45045 a^4 b c d^4+90090 a^3 b^2 c^2 d^3+50050 a^3 b^2 d^3 (c+d x)^2-128700 a^3 b^2 c d^3 (c+d x)-90090 a^2 b^3 c^3 d^2+193050 a^2 b^3 c^2 d^2 (c+d x)+40950 a^2 b^3 d^2 (c+d x)^3-150150 a^2 b^3 c d^2 (c+d x)^2+45045 a b^4 c^4 d-128700 a b^4 c^3 d (c+d x)+150150 a b^4 c^2 d (c+d x)^2+17325 a b^4 d (c+d x)^4-81900 a b^4 c d (c+d x)^3-9009 b^5 c^5+32175 b^5 c^4 (c+d x)-50050 b^5 c^3 (c+d x)^2+40950 b^5 c^2 (c+d x)^3+3003 b^5 (c+d x)^5-17325 b^5 c (c+d x)^4\right )}{45045 d^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.37, size = 418, normalized size = 2.65 \begin {gather*} \frac {2 \, {\left (3003 \, b^{5} d^{7} x^{7} - 256 \, b^{5} c^{7} + 1920 \, a b^{4} c^{6} d - 6240 \, a^{2} b^{3} c^{5} d^{2} + 11440 \, a^{3} b^{2} c^{4} d^{3} - 12870 \, a^{4} b c^{3} d^{4} + 9009 \, a^{5} c^{2} d^{5} + 231 \, {\left (16 \, b^{5} c d^{6} + 75 \, a b^{4} d^{7}\right )} x^{6} + 63 \, {\left (b^{5} c^{2} d^{5} + 350 \, a b^{4} c d^{6} + 650 \, a^{2} b^{3} d^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{3} d^{4} - 15 \, a b^{4} c^{2} d^{5} - 1560 \, a^{2} b^{3} c d^{6} - 1430 \, a^{3} b^{2} d^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} c^{4} d^{3} - 120 \, a b^{4} c^{3} d^{4} + 390 \, a^{2} b^{3} c^{2} d^{5} + 14300 \, a^{3} b^{2} c d^{6} + 6435 \, a^{4} b d^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{5} d^{2} - 240 \, a b^{4} c^{4} d^{3} + 780 \, a^{2} b^{3} c^{3} d^{4} - 1430 \, a^{3} b^{2} c^{2} d^{5} - 17160 \, a^{4} b c d^{6} - 3003 \, a^{5} d^{7}\right )} x^{2} + {\left (128 \, b^{5} c^{6} d - 960 \, a b^{4} c^{5} d^{2} + 3120 \, a^{2} b^{3} c^{4} d^{3} - 5720 \, a^{3} b^{2} c^{3} d^{4} + 6435 \, a^{4} b c^{2} d^{5} + 18018 \, a^{5} c d^{6}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.53, size = 1084, normalized size = 6.86
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 273, normalized size = 1.73 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (3003 b^{5} x^{5} d^{5}+17325 a \,b^{4} d^{5} x^{4}-2310 b^{5} c \,d^{4} x^{4}+40950 a^{2} b^{3} d^{5} x^{3}-12600 a \,b^{4} c \,d^{4} x^{3}+1680 b^{5} c^{2} d^{3} x^{3}+50050 a^{3} b^{2} d^{5} x^{2}-27300 a^{2} b^{3} c \,d^{4} x^{2}+8400 a \,b^{4} c^{2} d^{3} x^{2}-1120 b^{5} c^{3} d^{2} x^{2}+32175 a^{4} b \,d^{5} x -28600 a^{3} b^{2} c \,d^{4} x +15600 a^{2} b^{3} c^{2} d^{3} x -4800 a \,b^{4} c^{3} d^{2} x +640 b^{5} c^{4} d x +9009 a^{5} d^{5}-12870 a^{4} b c \,d^{4}+11440 a^{3} b^{2} c^{2} d^{3}-6240 a^{2} b^{3} c^{3} d^{2}+1920 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{45045 d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 259, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b^{5} - 17325 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 40950 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 50050 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 32175 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 9009 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{45045 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 137, normalized size = 0.87 \begin {gather*} \frac {2\,b^5\,{\left (c+d\,x\right )}^{15/2}}{15\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.42, size = 763, normalized size = 4.83
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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