3.377 \(\int x^2 \sqrt{a+b x} (A+B x) \, dx\)

Optimal. Leaf size=95 \[ \frac{2 a^2 (a+b x)^{3/2} (A b-a B)}{3 b^4}+\frac{2 (a+b x)^{7/2} (A b-3 a B)}{7 b^4}-\frac{2 a (a+b x)^{5/2} (2 A b-3 a B)}{5 b^4}+\frac{2 B (a+b x)^{9/2}}{9 b^4} \]

[Out]

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

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Rubi [A]  time = 0.124619, antiderivative size = 95, 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 \[ \frac{2 a^2 (a+b x)^{3/2} (A b-a B)}{3 b^4}+\frac{2 (a+b x)^{7/2} (A b-3 a B)}{7 b^4}-\frac{2 a (a+b x)^{5/2} (2 A b-3 a B)}{5 b^4}+\frac{2 B (a+b x)^{9/2}}{9 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[a + b*x]*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.1586, size = 92, normalized size = 0.97 \[ \frac{2 B \left (a + b x\right )^{\frac{9}{2}}}{9 b^{4}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{4}} - \frac{2 a \left (a + b x\right )^{\frac{5}{2}} \left (2 A b - 3 B a\right )}{5 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A b - 3 B a\right )}{7 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0693454, size = 65, normalized size = 0.68 \[ \frac{2 (a+b x)^{3/2} \left (-16 a^3 B+24 a^2 b (A+B x)-6 a b^2 x (6 A+5 B x)+5 b^3 x^2 (9 A+7 B x)\right )}{315 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[a + b*x]*(A + B*x),x]

[Out]

(2*(a + b*x)^(3/2)*(-16*a^3*B + 24*a^2*b*(A + B*x) - 6*a*b^2*x*(6*A + 5*B*x) + 5
*b^3*x^2*(9*A + 7*B*x)))/(315*b^4)

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Maple [A]  time = 0.009, size = 71, normalized size = 0.8 \[{\frac{70\,{b}^{3}B{x}^{3}+90\,A{x}^{2}{b}^{3}-60\,B{x}^{2}a{b}^{2}-72\,Axa{b}^{2}+48\,Bx{a}^{2}b+48\,A{a}^{2}b-32\,B{a}^{3}}{315\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x+A)*(b*x+a)^(1/2),x)

[Out]

2/315*(b*x+a)^(3/2)*(35*B*b^3*x^3+45*A*b^3*x^2-30*B*a*b^2*x^2-36*A*a*b^2*x+24*B*
a^2*b*x+24*A*a^2*b-16*B*a^3)/b^4

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Maxima [A]  time = 1.36407, size = 104, normalized size = 1.09 \[ \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 45 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}} + 63 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{5}{2}} - 105 \,{\left (B a^{3} - A a^{2} b\right )}{\left (b x + a\right )}^{\frac{3}{2}}\right )}}{315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)*x^2,x, algorithm="maxima")

[Out]

2/315*(35*(b*x + a)^(9/2)*B - 45*(3*B*a - A*b)*(b*x + a)^(7/2) + 63*(3*B*a^2 - 2
*A*a*b)*(b*x + a)^(5/2) - 105*(B*a^3 - A*a^2*b)*(b*x + a)^(3/2))/b^4

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Fricas [A]  time = 0.207722, size = 128, normalized size = 1.35 \[ \frac{2 \,{\left (35 \, B b^{4} x^{4} - 16 \, B a^{4} + 24 \, A a^{3} b + 5 \,{\left (B a b^{3} + 9 \, A b^{4}\right )} x^{3} - 3 \,{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{2} + 4 \,{\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)*x^2,x, algorithm="fricas")

[Out]

2/315*(35*B*b^4*x^4 - 16*B*a^4 + 24*A*a^3*b + 5*(B*a*b^3 + 9*A*b^4)*x^3 - 3*(2*B
*a^2*b^2 - 3*A*a*b^3)*x^2 + 4*(2*B*a^3*b - 3*A*a^2*b^2)*x)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 3.21127, size = 92, normalized size = 0.97 \[ \frac{2 \left (\frac{B \left (a + b x\right )^{\frac{9}{2}}}{9 b} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (A b - 3 B a\right )}{7 b} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (- 2 A a b + 3 B a^{2}\right )}{5 b} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (A a^{2} b - B a^{3}\right )}{3 b}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212718, size = 154, normalized size = 1.62 \[ \frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} A}{b^{14}} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} B}{b^{27}}\right )}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)*x^2,x, algorithm="giac")

[Out]

2/315*(3*(15*(b*x + a)^(7/2)*b^12 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/
2)*a^2*b^12)*A/b^14 + (35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2)*a*b^24 + 18
9*(b*x + a)^(5/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*B/b^27)/b