Optimal. Leaf size=122 \[ -\frac{2 a^3 (a+b x)^{3/2} (A b-a B)}{3 b^5}+\frac{2 a^2 (a+b x)^{5/2} (3 A b-4 a B)}{5 b^5}+\frac{2 (a+b x)^{9/2} (A b-4 a B)}{9 b^5}-\frac{6 a (a+b x)^{7/2} (A b-2 a B)}{7 b^5}+\frac{2 B (a+b x)^{11/2}}{11 b^5} \]
[Out]
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Rubi [A] time = 0.16052, antiderivative size = 122, 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 \[ -\frac{2 a^3 (a+b x)^{3/2} (A b-a B)}{3 b^5}+\frac{2 a^2 (a+b x)^{5/2} (3 A b-4 a B)}{5 b^5}+\frac{2 (a+b x)^{9/2} (A b-4 a B)}{9 b^5}-\frac{6 a (a+b x)^{7/2} (A b-2 a B)}{7 b^5}+\frac{2 B (a+b x)^{11/2}}{11 b^5} \]
Antiderivative was successfully verified.
[In] Int[x^3*Sqrt[a + b*x]*(A + B*x),x]
[Out]
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Rubi in Sympy [A] time = 22.4369, size = 119, normalized size = 0.98 \[ \frac{2 B \left (a + b x\right )^{\frac{11}{2}}}{11 b^{5}} - \frac{2 a^{3} \left (a + b x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{5}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{5}{2}} \left (3 A b - 4 B a\right )}{5 b^{5}} - \frac{6 a \left (a + b x\right )^{\frac{7}{2}} \left (A b - 2 B a\right )}{7 b^{5}} + \frac{2 \left (a + b x\right )^{\frac{9}{2}} \left (A b - 4 B a\right )}{9 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x+A)*(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0694437, size = 87, normalized size = 0.71 \[ \frac{2 (a+b x)^{3/2} \left (128 a^4 B-16 a^3 b (11 A+12 B x)+24 a^2 b^2 x (11 A+10 B x)-10 a b^3 x^2 (33 A+28 B x)+35 b^4 x^3 (11 A+9 B x)\right )}{3465 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*Sqrt[a + b*x]*(A + B*x),x]
[Out]
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Maple [A] time = 0.009, size = 95, normalized size = 0.8 \[ -{\frac{-630\,B{x}^{4}{b}^{4}-770\,A{b}^{4}{x}^{3}+560\,Ba{b}^{3}{x}^{3}+660\,Aa{b}^{3}{x}^{2}-480\,B{a}^{2}{b}^{2}{x}^{2}-528\,A{a}^{2}{b}^{2}x+384\,B{a}^{3}bx+352\,A{a}^{3}b-256\,B{a}^{4}}{3465\,{b}^{5}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x+A)*(b*x+a)^(1/2),x)
[Out]
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Maxima [A] time = 1.339, size = 135, normalized size = 1.11 \[ \frac{2 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} B - 385 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 1485 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{7}{2}} - 693 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 1155 \,{\left (B a^{4} - A a^{3} b\right )}{\left (b x + a\right )}^{\frac{3}{2}}\right )}}{3465 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(b*x + a)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204084, size = 161, normalized size = 1.32 \[ \frac{2 \,{\left (315 \, B b^{5} x^{5} + 128 \, B a^{5} - 176 \, A a^{4} b + 35 \,{\left (B a b^{4} + 11 \, A b^{5}\right )} x^{4} - 5 \,{\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{3465 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(b*x + a)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.49493, size = 121, normalized size = 0.99 \[ \frac{2 \left (\frac{B \left (a + b x\right )^{\frac{11}{2}}}{11 b} + \frac{\left (a + b x\right )^{\frac{9}{2}} \left (A b - 4 B a\right )}{9 b} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (- 3 A a b + 6 B a^{2}\right )}{7 b} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (3 A a^{2} b - 4 B a^{3}\right )}{5 b} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (- A a^{3} b + B a^{4}\right )}{3 b}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x+A)*(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210246, size = 194, normalized size = 1.59 \[ \frac{2 \,{\left (\frac{11 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} A}{b^{27}} + \frac{{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} B}{b^{44}}\right )}}{3465 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(b*x + a)*x^3,x, algorithm="giac")
[Out]