Optimal. Leaf size=129 \[ -\frac {8 b^3 (c+d x)^{11/2} (b c-a d)}{11 d^5}+\frac {4 b^2 (c+d x)^{9/2} (b c-a d)^2}{3 d^5}-\frac {8 b (c+d x)^{7/2} (b c-a d)^3}{7 d^5}+\frac {2 (c+d x)^{5/2} (b c-a d)^4}{5 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5} \]
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Rubi [A] time = 0.04, antiderivative size = 129, 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 {8 b^3 (c+d x)^{11/2} (b c-a d)}{11 d^5}+\frac {4 b^2 (c+d x)^{9/2} (b c-a d)^2}{3 d^5}-\frac {8 b (c+d x)^{7/2} (b c-a d)^3}{7 d^5}+\frac {2 (c+d x)^{5/2} (b c-a d)^4}{5 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^4 (c+d x)^{3/2} \, dx &=\int \left (\frac {(-b c+a d)^4 (c+d x)^{3/2}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{5/2}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{7/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{9/2}}{d^4}+\frac {b^4 (c+d x)^{11/2}}{d^4}\right ) \, dx\\ &=\frac {2 (b c-a d)^4 (c+d x)^{5/2}}{5 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{7/2}}{7 d^5}+\frac {4 b^2 (b c-a d)^2 (c+d x)^{9/2}}{3 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{11/2}}{11 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 101, normalized size = 0.78 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (-5460 b^3 (c+d x)^3 (b c-a d)+10010 b^2 (c+d x)^2 (b c-a d)^2-8580 b (c+d x) (b c-a d)^3+3003 (b c-a d)^4+1155 b^4 (c+d x)^4\right )}{15015 d^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 213, normalized size = 1.65 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (3003 a^4 d^4+8580 a^3 b d^3 (c+d x)-12012 a^3 b c d^3+18018 a^2 b^2 c^2 d^2+10010 a^2 b^2 d^2 (c+d x)^2-25740 a^2 b^2 c d^2 (c+d x)-12012 a b^3 c^3 d+25740 a b^3 c^2 d (c+d x)+5460 a b^3 d (c+d x)^3-20020 a b^3 c d (c+d x)^2+3003 b^4 c^4-8580 b^4 c^3 (c+d x)+10010 b^4 c^2 (c+d x)^2+1155 b^4 (c+d x)^4-5460 b^4 c (c+d x)^3\right )}{15015 d^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 311, normalized size = 2.41 \begin {gather*} \frac {2 \, {\left (1155 \, b^{4} d^{6} x^{6} + 128 \, b^{4} c^{6} - 832 \, a b^{3} c^{5} d + 2288 \, a^{2} b^{2} c^{4} d^{2} - 3432 \, a^{3} b c^{3} d^{3} + 3003 \, a^{4} c^{2} d^{4} + 210 \, {\left (7 \, b^{4} c d^{5} + 26 \, a b^{3} d^{6}\right )} x^{5} + 35 \, {\left (b^{4} c^{2} d^{4} + 208 \, a b^{3} c d^{5} + 286 \, a^{2} b^{2} d^{6}\right )} x^{4} - 20 \, {\left (2 \, b^{4} c^{3} d^{3} - 13 \, a b^{3} c^{2} d^{4} - 715 \, a^{2} b^{2} c d^{5} - 429 \, a^{3} b d^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{4} c^{4} d^{2} - 104 \, a b^{3} c^{3} d^{3} + 286 \, a^{2} b^{2} c^{2} d^{4} + 4576 \, a^{3} b c d^{5} + 1001 \, a^{4} d^{6}\right )} x^{2} - 2 \, {\left (32 \, b^{4} c^{5} d - 208 \, a b^{3} c^{4} d^{2} + 572 \, a^{2} b^{2} c^{3} d^{3} - 858 \, a^{3} b c^{2} d^{4} - 3003 \, a^{4} c d^{5}\right )} x\right )} \sqrt {d x + c}}{15015 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.40, size = 807, normalized size = 6.26
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 186, normalized size = 1.44 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (1155 b^{4} x^{4} d^{4}+5460 a \,b^{3} d^{4} x^{3}-840 b^{4} c \,d^{3} x^{3}+10010 a^{2} b^{2} d^{4} x^{2}-3640 a \,b^{3} c \,d^{3} x^{2}+560 b^{4} c^{2} d^{2} x^{2}+8580 a^{3} b \,d^{4} x -5720 a^{2} b^{2} c \,d^{3} x +2080 a \,b^{3} c^{2} d^{2} x -320 b^{4} c^{3} d x +3003 a^{4} d^{4}-3432 a^{3} b c \,d^{3}+2288 a^{2} b^{2} c^{2} d^{2}-832 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{15015 d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 181, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (1155 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{4} - 5460 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 8580 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 3003 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{15015 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (c+d\,x\right )}^{13/2}}{13\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}+\frac {4\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{3\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.09, size = 559, normalized size = 4.33 \begin {gather*} a^{4} c \left (\begin {cases} \sqrt {c} x & \text {for}\: d = 0 \\\frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{4} \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d} + \frac {8 a^{3} b c \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} + \frac {8 a^{3} b \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{2}} + \frac {12 a^{2} b^{2} c \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} + \frac {12 a^{2} b^{2} \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{3}} + \frac {8 a b^{3} c \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} + \frac {8 a b^{3} \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{4}} + \frac {2 b^{4} c \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} + \frac {2 b^{4} \left (- \frac {c^{5} \left (c + d x\right )^{\frac {3}{2}}}{3} + c^{4} \left (c + d x\right )^{\frac {5}{2}} - \frac {10 c^{3} \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {10 c^{2} \left (c + d x\right )^{\frac {9}{2}}}{9} - \frac {5 c \left (c + d x\right )^{\frac {11}{2}}}{11} + \frac {\left (c + d x\right )^{\frac {13}{2}}}{13}\right )}{d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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