3.376 \(\int x^3 \sqrt{a+b x} (A+B x) \, dx\)

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]

(-2*a^3*(A*b - a*B)*(a + b*x)^(3/2))/(3*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^
(5/2))/(5*b^5) - (6*a*(A*b - 2*a*B)*(a + b*x)^(7/2))/(7*b^5) + (2*(A*b - 4*a*B)*
(a + b*x)^(9/2))/(9*b^5) + (2*B*(a + b*x)^(11/2))/(11*b^5)

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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]

(-2*a^3*(A*b - a*B)*(a + b*x)^(3/2))/(3*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^
(5/2))/(5*b^5) - (6*a*(A*b - 2*a*B)*(a + b*x)^(7/2))/(7*b^5) + (2*(A*b - 4*a*B)*
(a + b*x)^(9/2))/(9*b^5) + (2*B*(a + b*x)^(11/2))/(11*b^5)

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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]

2*B*(a + b*x)**(11/2)/(11*b**5) - 2*a**3*(a + b*x)**(3/2)*(A*b - B*a)/(3*b**5) +
2*a**2*(a + b*x)**(5/2)*(3*A*b - 4*B*a)/(5*b**5) - 6*a*(a + b*x)**(7/2)*(A*b -
2*B*a)/(7*b**5) + 2*(a + b*x)**(9/2)*(A*b - 4*B*a)/(9*b**5)

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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]

(2*(a + b*x)^(3/2)*(128*a^4*B + 35*b^4*x^3*(11*A + 9*B*x) + 24*a^2*b^2*x*(11*A +
10*B*x) - 16*a^3*b*(11*A + 12*B*x) - 10*a*b^3*x^2*(33*A + 28*B*x)))/(3465*b^5)

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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]

-2/3465*(b*x+a)^(3/2)*(-315*B*b^4*x^4-385*A*b^4*x^3+280*B*a*b^3*x^3+330*A*a*b^3*
x^2-240*B*a^2*b^2*x^2-264*A*a^2*b^2*x+192*B*a^3*b*x+176*A*a^3*b-128*B*a^4)/b^5

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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]

2/3465*(315*(b*x + a)^(11/2)*B - 385*(4*B*a - A*b)*(b*x + a)^(9/2) + 1485*(2*B*a
^2 - A*a*b)*(b*x + a)^(7/2) - 693*(4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(5/2) + 1155*(
B*a^4 - A*a^3*b)*(b*x + a)^(3/2))/b^5

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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]

2/3465*(315*B*b^5*x^5 + 128*B*a^5 - 176*A*a^4*b + 35*(B*a*b^4 + 11*A*b^5)*x^4 -
5*(8*B*a^2*b^3 - 11*A*a*b^4)*x^3 + 6*(8*B*a^3*b^2 - 11*A*a^2*b^3)*x^2 - 8*(8*B*a
^4*b - 11*A*a^3*b^2)*x)*sqrt(b*x + a)/b^5

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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]

2*(B*(a + b*x)**(11/2)/(11*b) + (a + b*x)**(9/2)*(A*b - 4*B*a)/(9*b) + (a + b*x)
**(7/2)*(-3*A*a*b + 6*B*a**2)/(7*b) + (a + b*x)**(5/2)*(3*A*a**2*b - 4*B*a**3)/(
5*b) + (a + b*x)**(3/2)*(-A*a**3*b + B*a**4)/(3*b))/b**4

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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]

2/3465*(11*(35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2)*a*b^24 + 189*(b*x + a)
^(5/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*A/b^27 + (315*(b*x + a)^(11/2)*b
^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40 - 2772*(b*x +
a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*B/b^44)/b