Optimal. Leaf size=149 \[ \frac{4 a^2 (a+b x)^{5/2} (3 A b-5 a B)}{5 b^6}-\frac{2 a^3 (a+b x)^{3/2} (4 A b-5 a B)}{3 b^6}+\frac{2 a^4 \sqrt{a+b x} (A b-a B)}{b^6}+\frac{2 (a+b x)^{9/2} (A b-5 a B)}{9 b^6}-\frac{4 a (a+b x)^{7/2} (2 A b-5 a B)}{7 b^6}+\frac{2 B (a+b x)^{11/2}}{11 b^6} \]
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Rubi [A] time = 0.0571883, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{4 a^2 (a+b x)^{5/2} (3 A b-5 a B)}{5 b^6}-\frac{2 a^3 (a+b x)^{3/2} (4 A b-5 a B)}{3 b^6}+\frac{2 a^4 \sqrt{a+b x} (A b-a B)}{b^6}+\frac{2 (a+b x)^{9/2} (A b-5 a B)}{9 b^6}-\frac{4 a (a+b x)^{7/2} (2 A b-5 a B)}{7 b^6}+\frac{2 B (a+b x)^{11/2}}{11 b^6} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{x^4 (A+B x)}{\sqrt{a+b x}} \, dx &=\int \left (-\frac{a^4 (-A b+a B)}{b^5 \sqrt{a+b x}}+\frac{a^3 (-4 A b+5 a B) \sqrt{a+b x}}{b^5}-\frac{2 a^2 (-3 A b+5 a B) (a+b x)^{3/2}}{b^5}+\frac{2 a (-2 A b+5 a B) (a+b x)^{5/2}}{b^5}+\frac{(A b-5 a B) (a+b x)^{7/2}}{b^5}+\frac{B (a+b x)^{9/2}}{b^5}\right ) \, dx\\ &=\frac{2 a^4 (A b-a B) \sqrt{a+b x}}{b^6}-\frac{2 a^3 (4 A b-5 a B) (a+b x)^{3/2}}{3 b^6}+\frac{4 a^2 (3 A b-5 a B) (a+b x)^{5/2}}{5 b^6}-\frac{4 a (2 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac{2 (A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac{2 B (a+b x)^{11/2}}{11 b^6}\\ \end{align*}
Mathematica [A] time = 0.0781851, size = 106, normalized size = 0.71 \[ \frac{2 \sqrt{a+b x} \left (16 a^2 b^3 x^2 (33 A+25 B x)-32 a^3 b^2 x (22 A+15 B x)+128 a^4 b (11 A+5 B x)-1280 a^5 B-10 a b^4 x^3 (44 A+35 B x)+35 b^5 x^4 (11 A+9 B x)\right )}{3465 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 119, normalized size = 0.8 \begin{align*}{\frac{630\,{b}^{5}B{x}^{5}+770\,A{x}^{4}{b}^{5}-700\,B{x}^{4}a{b}^{4}-880\,A{x}^{3}a{b}^{4}+800\,B{x}^{3}{a}^{2}{b}^{3}+1056\,A{x}^{2}{a}^{2}{b}^{3}-960\,B{x}^{2}{a}^{3}{b}^{2}-1408\,{a}^{3}{b}^{2}Ax+1280\,{a}^{4}bBx+2816\,A{a}^{4}b-2560\,B{a}^{5}}{3465\,{b}^{6}}\sqrt{bx+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12021, size = 166, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} B - 385 \,{\left (5 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 990 \,{\left (5 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{7}{2}} - 1386 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )}{\left (b x + a\right )}^{\frac{3}{2}} - 3465 \,{\left (B a^{5} - A a^{4} b\right )} \sqrt{b x + a}\right )}}{3465 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41637, size = 289, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (315 \, B b^{5} x^{5} - 1280 \, B a^{5} + 1408 \, A a^{4} b - 35 \,{\left (10 \, B a b^{4} - 11 \, A b^{5}\right )} x^{4} + 40 \,{\left (10 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{3} - 48 \,{\left (10 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 64 \,{\left (10 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{3465 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 47.2333, size = 362, normalized size = 2.43 \begin{align*} \begin{cases} - \frac{\frac{2 A a \left (\frac{a^{4}}{\sqrt{a + b x}} + 4 a^{3} \sqrt{a + b x} - 2 a^{2} \left (a + b x\right )^{\frac{3}{2}} + \frac{4 a \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{4}} + \frac{2 A \left (- \frac{a^{5}}{\sqrt{a + b x}} - 5 a^{4} \sqrt{a + b x} + \frac{10 a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} - 2 a^{2} \left (a + b x\right )^{\frac{5}{2}} + \frac{5 a \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{4}} + \frac{2 B a \left (- \frac{a^{5}}{\sqrt{a + b x}} - 5 a^{4} \sqrt{a + b x} + \frac{10 a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} - 2 a^{2} \left (a + b x\right )^{\frac{5}{2}} + \frac{5 a \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{5}} + \frac{2 B \left (\frac{a^{6}}{\sqrt{a + b x}} + 6 a^{5} \sqrt{a + b x} - 5 a^{4} \left (a + b x\right )^{\frac{3}{2}} + 4 a^{3} \left (a + b x\right )^{\frac{5}{2}} - \frac{15 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{2 a \left (a + b x\right )^{\frac{9}{2}}}{3} - \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{5}}}{b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{5}}{5} + \frac{B x^{6}}{6}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15269, size = 192, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (\frac{11 \,{\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}{b^{4}} + \frac{5 \,{\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 )} B}{b^{5}}\right )}}{3465 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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