Optimal. Leaf size=42 \[ \frac{2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (a+b x)^{9/2}}{9 b^2} \]
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Rubi [A] time = 0.0141924, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (a+b x)^{9/2}}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^{5/2} (A+B x) \, dx &=\int \left (\frac{(A b-a B) (a+b x)^{5/2}}{b}+\frac{B (a+b x)^{7/2}}{b}\right ) \, dx\\ &=\frac{2 (A b-a B) (a+b x)^{7/2}}{7 b^2}+\frac{2 B (a+b x)^{9/2}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.0215187, size = 30, normalized size = 0.71 \[ \frac{2 (a+b x)^{7/2} (-2 a B+9 A b+7 b B x)}{63 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 0.6 \begin{align*}{\frac{14\,bBx+18\,Ab-4\,Ba}{63\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52742, size = 45, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (7 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 9 \,{\left (B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{63 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.34175, size = 205, normalized size = 4.88 \begin{align*} \frac{2 \,{\left (7 \, B b^{4} x^{4} - 2 \, B a^{4} + 9 \, A a^{3} b +{\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{2} +{\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{63 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.38924, size = 194, normalized size = 4.62 \begin{align*} \begin{cases} \frac{2 A a^{3} \sqrt{a + b x}}{7 b} + \frac{6 A a^{2} x \sqrt{a + b x}}{7} + \frac{6 A a b x^{2} \sqrt{a + b x}}{7} + \frac{2 A b^{2} x^{3} \sqrt{a + b x}}{7} - \frac{4 B a^{4} \sqrt{a + b x}}{63 b^{2}} + \frac{2 B a^{3} x \sqrt{a + b x}}{63 b} + \frac{10 B a^{2} x^{2} \sqrt{a + b x}}{21} + \frac{38 B a b x^{3} \sqrt{a + b x}}{63} + \frac{2 B b^{2} x^{4} \sqrt{a + b x}}{9} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (A x + \frac{B x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28132, size = 263, normalized size = 6.26 \begin{align*} \frac{2 \,{\left (105 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} + 42 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A a + \frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} B a^{2}}{b} + 3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} A + \frac{6 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} B a}{b} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} B}{b}\right )}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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