Optimal. Leaf size=151 \[ \frac{4 a^2 (a+b x)^{11/2} (3 A b-5 a B)}{11 b^6}-\frac{2 a^3 (a+b x)^{9/2} (4 A b-5 a B)}{9 b^6}+\frac{2 a^4 (a+b x)^{7/2} (A b-a B)}{7 b^6}+\frac{2 (a+b x)^{15/2} (A b-5 a B)}{15 b^6}-\frac{4 a (a+b x)^{13/2} (2 A b-5 a B)}{13 b^6}+\frac{2 B (a+b x)^{17/2}}{17 b^6} \]
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Rubi [A] time = 0.0602927, antiderivative size = 151, 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} \[ \frac{4 a^2 (a+b x)^{11/2} (3 A b-5 a B)}{11 b^6}-\frac{2 a^3 (a+b x)^{9/2} (4 A b-5 a B)}{9 b^6}+\frac{2 a^4 (a+b x)^{7/2} (A b-a B)}{7 b^6}+\frac{2 (a+b x)^{15/2} (A b-5 a B)}{15 b^6}-\frac{4 a (a+b x)^{13/2} (2 A b-5 a B)}{13 b^6}+\frac{2 B (a+b x)^{17/2}}{17 b^6} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int x^4 (a+b x)^{5/2} (A+B x) \, dx &=\int \left (-\frac{a^4 (-A b+a B) (a+b x)^{5/2}}{b^5}+\frac{a^3 (-4 A b+5 a B) (a+b x)^{7/2}}{b^5}-\frac{2 a^2 (-3 A b+5 a B) (a+b x)^{9/2}}{b^5}+\frac{2 a (-2 A b+5 a B) (a+b x)^{11/2}}{b^5}+\frac{(A b-5 a B) (a+b x)^{13/2}}{b^5}+\frac{B (a+b x)^{15/2}}{b^5}\right ) \, dx\\ &=\frac{2 a^4 (A b-a B) (a+b x)^{7/2}}{7 b^6}-\frac{2 a^3 (4 A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac{4 a^2 (3 A b-5 a B) (a+b x)^{11/2}}{11 b^6}-\frac{4 a (2 A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac{2 (A b-5 a B) (a+b x)^{15/2}}{15 b^6}+\frac{2 B (a+b x)^{17/2}}{17 b^6}\\ \end{align*}
Mathematica [A] time = 0.0992304, size = 106, normalized size = 0.7 \[ \frac{2 (a+b x)^{7/2} \left (336 a^2 b^3 x^2 (51 A+55 B x)-224 a^3 b^2 x (34 A+45 B x)+128 a^4 b (17 A+35 B x)-1280 a^5 B-462 a b^4 x^3 (68 A+65 B x)+3003 b^5 x^4 (17 A+15 B x)\right )}{765765 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 119, normalized size = 0.8 \begin{align*}{\frac{90090\,{b}^{5}B{x}^{5}+102102\,A{x}^{4}{b}^{5}-60060\,B{x}^{4}a{b}^{4}-62832\,A{x}^{3}a{b}^{4}+36960\,B{x}^{3}{a}^{2}{b}^{3}+34272\,A{x}^{2}{a}^{2}{b}^{3}-20160\,B{x}^{2}{a}^{3}{b}^{2}-15232\,{a}^{3}{b}^{2}Ax+8960\,{a}^{4}bBx+4352\,A{a}^{4}b-2560\,B{a}^{5}}{765765\,{b}^{6}} \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 = 1.01982, size = 166, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (45045 \,{\left (b x + a\right )}^{\frac{17}{2}} B - 51051 \,{\left (5 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{15}{2}} + 117810 \,{\left (5 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{13}{2}} - 139230 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )}{\left (b x + a\right )}^{\frac{9}{2}} - 109395 \,{\left (B a^{5} - A a^{4} b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{765765 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21601, size = 460, normalized size = 3.05 \begin{align*} \frac{2 \,{\left (45045 \, B b^{8} x^{8} - 1280 \, B a^{8} + 2176 \, A a^{7} b + 3003 \,{\left (35 \, B a b^{7} + 17 \, A b^{8}\right )} x^{7} + 231 \,{\left (275 \, B a^{2} b^{6} + 527 \, A a b^{7}\right )} x^{6} + 63 \,{\left (5 \, B a^{3} b^{5} + 1207 \, A a^{2} b^{6}\right )} x^{5} - 35 \,{\left (10 \, B a^{4} b^{4} - 17 \, A a^{3} b^{5}\right )} x^{4} + 40 \,{\left (10 \, B a^{5} b^{3} - 17 \, A a^{4} b^{4}\right )} x^{3} - 48 \,{\left (10 \, B a^{6} b^{2} - 17 \, A a^{5} b^{3}\right )} x^{2} + 64 \,{\left (10 \, B a^{7} b - 17 \, A a^{6} b^{2}\right )} x\right )} \sqrt{b x + a}}{765765 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 20.6639, size = 586, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22163, size = 667, normalized size = 4.42 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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