Optimal. Leaf size=122 \[ -\frac{2 a^3 (a+b x)^{7/2} (A b-a B)}{7 b^5}+\frac{2 a^2 (a+b x)^{9/2} (3 A b-4 a B)}{9 b^5}+\frac{2 (a+b x)^{13/2} (A b-4 a B)}{13 b^5}-\frac{6 a (a+b x)^{11/2} (A b-2 a B)}{11 b^5}+\frac{2 B (a+b x)^{15/2}}{15 b^5} \]
[Out]
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Rubi [A] time = 0.156884, 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 \[ -\frac{2 a^3 (a+b x)^{7/2} (A b-a B)}{7 b^5}+\frac{2 a^2 (a+b x)^{9/2} (3 A b-4 a B)}{9 b^5}+\frac{2 (a+b x)^{13/2} (A b-4 a B)}{13 b^5}-\frac{6 a (a+b x)^{11/2} (A b-2 a B)}{11 b^5}+\frac{2 B (a+b x)^{15/2}}{15 b^5} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x)^(5/2)*(A + B*x),x]
[Out]
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Rubi in Sympy [A] time = 27.1526, size = 119, normalized size = 0.98 \[ \frac{2 B \left (a + b x\right )^{\frac{15}{2}}}{15 b^{5}} - \frac{2 a^{3} \left (a + b x\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{5}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{9}{2}} \left (3 A b - 4 B a\right )}{9 b^{5}} - \frac{6 a \left (a + b x\right )^{\frac{11}{2}} \left (A b - 2 B a\right )}{11 b^{5}} + \frac{2 \left (a + b x\right )^{\frac{13}{2}} \left (A b - 4 B a\right )}{13 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x+a)**(5/2)*(B*x+A),x)
[Out]
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Mathematica [A] time = 0.0945166, size = 87, normalized size = 0.71 \[ \frac{2 (a+b x)^{7/2} \left (128 a^4 B-16 a^3 b (15 A+28 B x)+168 a^2 b^2 x (5 A+6 B x)-42 a b^3 x^2 (45 A+44 B x)+231 b^4 x^3 (15 A+13 B x)\right )}{45045 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(a + b*x)^(5/2)*(A + B*x),x]
[Out]
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Maple [A] time = 0.009, size = 95, normalized size = 0.8 \[ -{\frac{-6006\,B{x}^{4}{b}^{4}-6930\,A{b}^{4}{x}^{3}+3696\,Ba{b}^{3}{x}^{3}+3780\,Aa{b}^{3}{x}^{2}-2016\,B{a}^{2}{b}^{2}{x}^{2}-1680\,A{a}^{2}{b}^{2}x+896\,B{a}^{3}bx+480\,A{a}^{3}b-256\,B{a}^{4}}{45045\,{b}^{5}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x+a)^(5/2)*(B*x+A),x)
[Out]
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Maxima [A] time = 1.36065, size = 135, normalized size = 1.11 \[ \frac{2 \,{\left (3003 \,{\left (b x + a\right )}^{\frac{15}{2}} B - 3465 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{13}{2}} + 12285 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{11}{2}} - 5005 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 6435 \,{\left (B a^{4} - A a^{3} b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{45045 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207357, size = 225, normalized size = 1.84 \[ \frac{2 \,{\left (3003 \, B b^{7} x^{7} + 128 \, B a^{7} - 240 \, A a^{6} b + 231 \,{\left (31 \, B a b^{6} + 15 \, A b^{7}\right )} x^{6} + 63 \,{\left (71 \, B a^{2} b^{5} + 135 \, A a b^{6}\right )} x^{5} + 35 \,{\left (B a^{3} b^{4} + 159 \, A a^{2} b^{5}\right )} x^{4} - 5 \,{\left (8 \, B a^{4} b^{3} - 15 \, A a^{3} b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{6} b - 15 \, A a^{5} b^{2}\right )} x\right )} \sqrt{b x + a}}{45045 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.728, size = 496, normalized size = 4.07 \[ \frac{2 A a^{2} \left (- \frac{a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{3 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{4}} + \frac{4 A a \left (\frac{a^{4} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{4 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{4}} + \frac{2 A \left (- \frac{a^{5} \left (a + b x\right )^{\frac{3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac{5}{2}} - \frac{10 a^{3} \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{10 a^{2} \left (a + b x\right )^{\frac{9}{2}}}{9} - \frac{5 a \left (a + b x\right )^{\frac{11}{2}}}{11} + \frac{\left (a + b x\right )^{\frac{13}{2}}}{13}\right )}{b^{4}} + \frac{2 B a^{2} \left (\frac{a^{4} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{4 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{5}} + \frac{4 B a \left (- \frac{a^{5} \left (a + b x\right )^{\frac{3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac{5}{2}} - \frac{10 a^{3} \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{10 a^{2} \left (a + b x\right )^{\frac{9}{2}}}{9} - \frac{5 a \left (a + b x\right )^{\frac{11}{2}}}{11} + \frac{\left (a + b x\right )^{\frac{13}{2}}}{13}\right )}{b^{5}} + \frac{2 B \left (\frac{a^{6} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{6 a^{5} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{15 a^{4} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{20 a^{3} \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{15 a^{2} \left (a + b x\right )^{\frac{11}{2}}}{11} - \frac{6 a \left (a + b x\right )^{\frac{13}{2}}}{13} + \frac{\left (a + b x\right )^{\frac{15}{2}}}{15}\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x+a)**(5/2)*(B*x+A),x)
[Out]
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GIAC/XCAS [A] time = 0.24979, size = 702, normalized size = 5.75 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)*x^3,x, algorithm="giac")
[Out]