Optimal. Leaf size=72 \[ -\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4} \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^3 (a+b x)^{5/2} \, dx &=\int \left (-\frac {a^3 (a+b x)^{5/2}}{b^3}+\frac {3 a^2 (a+b x)^{7/2}}{b^3}-\frac {3 a (a+b x)^{9/2}}{b^3}+\frac {(a+b x)^{11/2}}{b^3}\right ) \, dx\\ &=-\frac {2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac {2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac {6 a (a+b x)^{11/2}}{11 b^4}+\frac {2 (a+b x)^{13/2}}{13 b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.64 \[ \frac {2 (a+b x)^{7/2} \left (-16 a^3+56 a^2 b x-126 a b^2 x^2+231 b^3 x^3\right )}{3003 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 75, normalized size = 1.04 \[ \frac {2 \, {\left (231 \, b^{6} x^{6} + 567 \, a b^{5} x^{5} + 371 \, a^{2} b^{4} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{4} b^{2} x^{2} + 8 \, a^{5} b x - 16 \, a^{6}\right )} \sqrt {b x + a}}{3003 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 281, normalized size = 3.90 \[ \frac {2 \, {\left (\frac {429 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{3}}{b^{3}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a^{2}}{b^{3}} + \frac {65 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a}{b^{3}} + \frac {5 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )}}{b^{3}}\right )}}{15015 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.60 \[ -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-231 b^{3} x^{3}+126 a \,b^{2} x^{2}-56 a^{2} b x +16 a^{3}\right )}{3003 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 56, normalized size = 0.78 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {13}{2}}}{13 \, b^{4}} - \frac {6 \, {\left (b x + a\right )}^{\frac {11}{2}} a}{11 \, b^{4}} + \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2}}{3 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 56, normalized size = 0.78 \[ \frac {2\,{\left (a+b\,x\right )}^{13/2}}{13\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{9/2}}{3\,b^4}-\frac {6\,a\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.47, size = 146, normalized size = 2.03 \[ \begin {cases} - \frac {32 a^{6} \sqrt {a + b x}}{3003 b^{4}} + \frac {16 a^{5} x \sqrt {a + b x}}{3003 b^{3}} - \frac {4 a^{4} x^{2} \sqrt {a + b x}}{1001 b^{2}} + \frac {10 a^{3} x^{3} \sqrt {a + b x}}{3003 b} + \frac {106 a^{2} x^{4} \sqrt {a + b x}}{429} + \frac {54 a b x^{5} \sqrt {a + b x}}{143} + \frac {2 b^{2} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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