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

Optimal. Leaf size=67 \[ \frac{2 (a+b x)^{7/2} (A b-2 a B)}{7 b^3}-\frac{2 a (a+b x)^{5/2} (A b-a B)}{5 b^3}+\frac{2 B (a+b x)^{9/2}}{9 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0822411, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{2 (a+b x)^{7/2} (A b-2 a B)}{7 b^3}-\frac{2 a (a+b x)^{5/2} (A b-a B)}{5 b^3}+\frac{2 B (a+b x)^{9/2}}{9 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9073, size = 63, normalized size = 0.94 \[ \frac{2 B \left (a + b x\right )^{\frac{9}{2}}}{9 b^{3}} - \frac{2 a \left (a + b x\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A b - 2 B a\right )}{7 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0526935, size = 49, normalized size = 0.73 \[ \frac{2 (a+b x)^{5/2} \left (8 a^2 B-2 a b (9 A+10 B x)+5 b^2 x (9 A+7 B x)\right )}{315 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(5/2)*(8*a^2*B + 5*b^2*x*(9*A + 7*B*x) - 2*a*b*(9*A + 10*B*x)))/(31
5*b^3)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.7 \[ -{\frac{-70\,{b}^{2}B{x}^{2}-90\,Ax{b}^{2}+40\,Bxab+36\,Aab-16\,B{a}^{2}}{315\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(3/2)*(B*x+A),x)

[Out]

-2/315*(b*x+a)^(5/2)*(-35*B*b^2*x^2-45*A*b^2*x+20*B*a*b*x+18*A*a*b-8*B*a^2)/b^3

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Maxima [A]  time = 1.37418, size = 73, normalized size = 1.09 \[ \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 45 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}} + 63 \,{\left (B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{5}{2}}\right )}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205634, size = 128, normalized size = 1.91 \[ \frac{2 \,{\left (35 \, B b^{4} x^{4} + 8 \, B a^{4} - 18 \, A a^{3} b + 5 \,{\left (10 \, B a b^{3} + 9 \, A b^{4}\right )} x^{3} + 3 \,{\left (B a^{2} b^{2} + 24 \, A a b^{3}\right )} x^{2} -{\left (4 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*b^4*x^4 + 8*B*a^4 - 18*A*a^3*b + 5*(10*B*a*b^3 + 9*A*b^4)*x^3 + 3*(B
*a^2*b^2 + 24*A*a*b^3)*x^2 - (4*B*a^3*b - 9*A*a^2*b^2)*x)*sqrt(b*x + a)/b^3

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Sympy [A]  time = 4.68946, size = 178, normalized size = 2.66 \[ \frac{2 A a \left (- \frac{a \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{2}} + \frac{2 A \left (\frac{a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{2 a \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{2}} + \frac{2 B a \left (\frac{a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{2 a \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{3}} + \frac{2 B \left (- \frac{a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{3 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211581, size = 255, normalized size = 3.81 \[ \frac{2 \,{\left (\frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A a}{b} + \frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} B a}{b^{14}} + \frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} A}{b^{13}} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} B}{b^{26}}\right )}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(21*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*A*a/b + 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)*B*a/b^14 + 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^1
2)*A/b^13 + (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/b^26)/b