3.388 \(\int x^2 (a+b x)^{3/2} (A+B x) \, dx\)

Optimal. Leaf size=95 \[ \frac{2 a^2 (a+b x)^{5/2} (A b-a B)}{5 b^4}+\frac{2 (a+b x)^{9/2} (A b-3 a B)}{9 b^4}-\frac{2 a (a+b x)^{7/2} (2 A b-3 a B)}{7 b^4}+\frac{2 B (a+b x)^{11/2}}{11 b^4} \]

[Out]

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

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Rubi [A]  time = 0.123772, 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 \[ \frac{2 a^2 (a+b x)^{5/2} (A b-a B)}{5 b^4}+\frac{2 (a+b x)^{9/2} (A b-3 a B)}{9 b^4}-\frac{2 a (a+b x)^{7/2} (2 A b-3 a B)}{7 b^4}+\frac{2 B (a+b x)^{11/2}}{11 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.8154, size = 92, normalized size = 0.97 \[ \frac{2 B \left (a + b x\right )^{\frac{11}{2}}}{11 b^{4}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{4}} - \frac{2 a \left (a + b x\right )^{\frac{7}{2}} \left (2 A b - 3 B a\right )}{7 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{9}{2}} \left (A b - 3 B a\right )}{9 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

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

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Mathematica [A]  time = 0.0821374, size = 68, normalized size = 0.72 \[ \frac{2 (a+b x)^{5/2} \left (-48 a^3 B+8 a^2 b (11 A+15 B x)-10 a b^2 x (22 A+21 B x)+35 b^3 x^2 (11 A+9 B x)\right )}{3465 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(5/2)*(-48*a^3*B + 35*b^3*x^2*(11*A + 9*B*x) + 8*a^2*b*(11*A + 15*B
*x) - 10*a*b^2*x*(22*A + 21*B*x)))/(3465*b^4)

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Maple [A]  time = 0.009, size = 71, normalized size = 0.8 \[{\frac{630\,{b}^{3}B{x}^{3}+770\,A{x}^{2}{b}^{3}-420\,B{x}^{2}a{b}^{2}-440\,Axa{b}^{2}+240\,Bx{a}^{2}b+176\,A{a}^{2}b-96\,B{a}^{3}}{3465\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(3/2)*(B*x+A),x)

[Out]

2/3465*(b*x+a)^(5/2)*(315*B*b^3*x^3+385*A*b^3*x^2-210*B*a*b^2*x^2-220*A*a*b^2*x+
120*B*a^2*b*x+88*A*a^2*b-48*B*a^3)/b^4

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Maxima [A]  time = 1.45899, size = 104, normalized size = 1.09 \[ \frac{2 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} B - 385 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{7}{2}} - 693 \,{\left (B a^{3} - A a^{2} b\right )}{\left (b x + a\right )}^{\frac{5}{2}}\right )}}{3465 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)*x^2,x, algorithm="maxima")

[Out]

2/3465*(315*(b*x + a)^(11/2)*B - 385*(3*B*a - A*b)*(b*x + a)^(9/2) + 495*(3*B*a^
2 - 2*A*a*b)*(b*x + a)^(7/2) - 693*(B*a^3 - A*a^2*b)*(b*x + a)^(5/2))/b^4

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Fricas [A]  time = 0.203761, size = 162, normalized size = 1.71 \[ \frac{2 \,{\left (315 \, B b^{5} x^{5} - 48 \, B a^{5} + 88 \, A a^{4} b + 35 \,{\left (12 \, B a b^{4} + 11 \, A b^{5}\right )} x^{4} + 5 \,{\left (3 \, B a^{2} b^{3} + 110 \, A a b^{4}\right )} x^{3} - 3 \,{\left (6 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (6 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{3465 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)*x^2,x, algorithm="fricas")

[Out]

2/3465*(315*B*b^5*x^5 - 48*B*a^5 + 88*A*a^4*b + 35*(12*B*a*b^4 + 11*A*b^5)*x^4 +
5*(3*B*a^2*b^3 + 110*A*a*b^4)*x^3 - 3*(6*B*a^3*b^2 - 11*A*a^2*b^3)*x^2 + 4*(6*B
*a^4*b - 11*A*a^3*b^2)*x)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 5.13319, size = 240, normalized size = 2.53 \[ \frac{2 A a \left (\frac{a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{2 a \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{3}} + \frac{2 A \left (- \frac{a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{3 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{3}} + \frac{2 B a \left (- \frac{a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{3 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{4}} + \frac{2 B \left (\frac{a^{4} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{4 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214493, size = 344, normalized size = 3.62 \[ \frac{2 \,{\left (\frac{33 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} A a}{b^{14}} + \frac{11 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} B a}{b^{27}} + \frac{11 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} A}{b^{26}} + \frac{{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} B}{b^{43}}\right )}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)*x^2,x, algorithm="giac")

[Out]

2/3465*(33*(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*a/b^14 + 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)*B*a/b^27 + 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^26 + (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^43)/b