Optimal. Leaf size=71 \[ -\frac {4 b (c+d x)^{9/2} (b c-a d)}{9 d^3}+\frac {2 (c+d x)^{7/2} (b c-a d)^2}{7 d^3}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^3} \]
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Rubi [A] time = 0.02, antiderivative size = 71, 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} \[ -\frac {4 b (c+d x)^{9/2} (b c-a d)}{9 d^3}+\frac {2 (c+d x)^{7/2} (b c-a d)^2}{7 d^3}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^2 (c+d x)^{5/2} \, dx &=\int \left (\frac {(-b c+a d)^2 (c+d x)^{5/2}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{7/2}}{d^2}+\frac {b^2 (c+d x)^{9/2}}{d^2}\right ) \, dx\\ &=\frac {2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^3}-\frac {4 b (b c-a d) (c+d x)^{9/2}}{9 d^3}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.86 \[ \frac {2 (c+d x)^{7/2} \left (99 a^2 d^2+22 a b d (7 d x-2 c)+b^2 \left (8 c^2-28 c d x+63 d^2 x^2\right )\right )}{693 d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 174, normalized size = 2.45 \[ \frac {2 \, {\left (63 \, b^{2} d^{5} x^{5} + 8 \, b^{2} c^{5} - 44 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} + 7 \, {\left (23 \, b^{2} c d^{4} + 22 \, a b d^{5}\right )} x^{4} + {\left (113 \, b^{2} c^{2} d^{3} + 418 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{3} + 3 \, {\left (b^{2} c^{3} d^{2} + 110 \, a b c^{2} d^{3} + 99 \, a^{2} c d^{4}\right )} x^{2} - {\left (4 \, b^{2} c^{4} d - 22 \, a b c^{3} d^{2} - 297 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{693 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.76, size = 558, normalized size = 7.86 \[ \frac {2 \, {\left (3465 \, \sqrt {d x + c} a^{2} c^{3} + 3465 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} c^{2} + \frac {2310 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a b c^{3}}{d} + 693 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{2} c + \frac {231 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} b^{2} c^{3}}{d^{2}} + \frac {1386 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b c^{2}}{d} + 99 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{2} + \frac {297 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{2} c^{2}}{d^{2}} + \frac {594 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a b c}{d} + \frac {33 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b^{2} c}{d^{2}} + \frac {22 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b}{d} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{2}}{d^{2}}\right )}}{3465 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.89 \[ \frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (63 b^{2} x^{2} d^{2}+154 a b \,d^{2} x -28 b^{2} c d x +99 a^{2} d^{2}-44 a b c d +8 b^{2} c^{2}\right )}{693 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 68, normalized size = 0.96 \[ \frac {2 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{2} - 154 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 99 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{693 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 68, normalized size = 0.96 \[ \frac {2\,{\left (c+d\,x\right )}^{7/2}\,\left (63\,b^2\,{\left (c+d\,x\right )}^2+99\,a^2\,d^2+99\,b^2\,c^2-154\,b^2\,c\,\left (c+d\,x\right )+154\,a\,b\,d\,\left (c+d\,x\right )-198\,a\,b\,c\,d\right )}{693\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.58, size = 355, normalized size = 5.00 \[ \begin {cases} \frac {2 a^{2} c^{3} \sqrt {c + d x}}{7 d} + \frac {6 a^{2} c^{2} x \sqrt {c + d x}}{7} + \frac {6 a^{2} c d x^{2} \sqrt {c + d x}}{7} + \frac {2 a^{2} d^{2} x^{3} \sqrt {c + d x}}{7} - \frac {8 a b c^{4} \sqrt {c + d x}}{63 d^{2}} + \frac {4 a b c^{3} x \sqrt {c + d x}}{63 d} + \frac {20 a b c^{2} x^{2} \sqrt {c + d x}}{21} + \frac {76 a b c d x^{3} \sqrt {c + d x}}{63} + \frac {4 a b d^{2} x^{4} \sqrt {c + d x}}{9} + \frac {16 b^{2} c^{5} \sqrt {c + d x}}{693 d^{3}} - \frac {8 b^{2} c^{4} x \sqrt {c + d x}}{693 d^{2}} + \frac {2 b^{2} c^{3} x^{2} \sqrt {c + d x}}{231 d} + \frac {226 b^{2} c^{2} x^{3} \sqrt {c + d x}}{693} + \frac {46 b^{2} c d x^{4} \sqrt {c + d x}}{99} + \frac {2 b^{2} d^{2} x^{5} \sqrt {c + d x}}{11} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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