3.378 \(\int x \sqrt{a+b x} (A+B x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0815349, 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 \[ \frac{2 (a+b x)^{5/2} (A b-2 a B)}{5 b^3}-\frac{2 a (a+b x)^{3/2} (A b-a B)}{3 b^3}+\frac{2 B (a+b x)^{7/2}}{7 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.807, size = 63, normalized size = 0.94 \[ \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3}} - \frac{2 a \left (a + b x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A b - 2 B a\right )}{5 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.042855, size = 49, normalized size = 0.73 \[ \frac{2 (a+b x)^{3/2} \left (8 a^2 B-2 a b (7 A+6 B x)+3 b^2 x (7 A+5 B x)\right )}{105 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(3/2)*(8*a^2*B + 3*b^2*x*(7*A + 5*B*x) - 2*a*b*(7*A + 6*B*x)))/(105
*b^3)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.7 \[ -{\frac{-30\,{b}^{2}B{x}^{2}-42\,Ax{b}^{2}+24\,Bxab+28\,Aab-16\,B{a}^{2}}{105\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)*(b*x+a)^(1/2),x)

[Out]

-2/105*(b*x+a)^(3/2)*(-15*B*b^2*x^2-21*A*b^2*x+12*B*a*b*x+14*A*a*b-8*B*a^2)/b^3

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Maxima [A]  time = 1.34578, size = 73, normalized size = 1.09 \[ \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} B - 21 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 35 \,{\left (B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{3}{2}}\right )}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*(b*x + a)^(7/2)*B - 21*(2*B*a - A*b)*(b*x + a)^(5/2) + 35*(B*a^2 - A*a
*b)*(b*x + a)^(3/2))/b^3

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Fricas [A]  time = 0.207886, size = 96, normalized size = 1.43 \[ \frac{2 \,{\left (15 \, B b^{3} x^{3} + 8 \, B a^{3} - 14 \, A a^{2} b + 3 \,{\left (B a b^{2} + 7 \, A b^{3}\right )} x^{2} -{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*B*b^3*x^3 + 8*B*a^3 - 14*A*a^2*b + 3*(B*a*b^2 + 7*A*b^3)*x^2 - (4*B*a^
2*b - 7*A*a*b^2)*x)*sqrt(b*x + a)/b^3

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Sympy [A]  time = 2.99707, size = 63, normalized size = 0.94 \[ \frac{2 \left (\frac{B \left (a + b x\right )^{\frac{7}{2}}}{7 b} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (A b - 2 B a\right )}{5 b} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (- A a b + B a^{2}\right )}{3 b}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.237735, size = 105, normalized size = 1.57 \[ \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A}{b} + \frac{{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} B}{b^{14}}\right )}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(7*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*A/b + (15*(b*x + a)^(7/2)*b^1
2 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/2)*a^2*b^12)*B/b^14)/b