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

Optimal. Leaf size=93 \[ \frac{2 a^2 \sqrt{a+b x} (A b-a B)}{b^4}+\frac{2 (a+b x)^{5/2} (A b-3 a B)}{5 b^4}-\frac{2 a (a+b x)^{3/2} (2 A b-3 a B)}{3 b^4}+\frac{2 B (a+b x)^{7/2}}{7 b^4} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.121947, antiderivative size = 93, 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 \[ \frac{2 a^2 \sqrt{a+b x} (A b-a B)}{b^4}+\frac{2 (a+b x)^{5/2} (A b-3 a B)}{5 b^4}-\frac{2 a (a+b x)^{3/2} (2 A b-3 a B)}{3 b^4}+\frac{2 B (a+b x)^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.4181, size = 90, normalized size = 0.97 \[ \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{2 a^{2} \sqrt{a + b x} \left (A b - B a\right )}{b^{4}} - \frac{2 a \left (a + b x\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{3 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A b - 3 B a\right )}{5 b^{4}} \]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0573486, size = 68, normalized size = 0.73 \[ \frac{2 \sqrt{a+b x} \left (-48 a^3 B+8 a^2 b (7 A+3 B x)-2 a b^2 x (14 A+9 B x)+3 b^3 x^2 (7 A+5 B x)\right )}{105 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x]*(-48*a^3*B + 8*a^2*b*(7*A + 3*B*x) + 3*b^3*x^2*(7*A + 5*B*x) -
2*a*b^2*x*(14*A + 9*B*x)))/(105*b^4)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 71, normalized size = 0.8 \[{\frac{30\,{b}^{3}B{x}^{3}+42\,A{x}^{2}{b}^{3}-36\,B{x}^{2}a{b}^{2}-56\,Axa{b}^{2}+48\,Bx{a}^{2}b+112\,A{a}^{2}b-96\,B{a}^{3}}{105\,{b}^{4}}\sqrt{bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(b*x+a)^(1/2)*(15*B*b^3*x^3+21*A*b^3*x^2-18*B*a*b^2*x^2-28*A*a*b^2*x+24*B*
a^2*b*x+56*A*a^2*b-48*B*a^3)/b^4

_______________________________________________________________________________________

Maxima [A]  time = 1.35205, size = 104, normalized size = 1.12 \[ \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} B - 21 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{3}{2}} - 105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{b x + a}\right )}}{105 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*(b*x + a)^(7/2)*B - 21*(3*B*a - A*b)*(b*x + a)^(5/2) + 35*(3*B*a^2 - 2
*A*a*b)*(b*x + a)^(3/2) - 105*(B*a^3 - A*a^2*b)*sqrt(b*x + a))/b^4

_______________________________________________________________________________________

Fricas [A]  time = 0.204595, size = 97, normalized size = 1.04 \[ \frac{2 \,{\left (15 \, B b^{3} x^{3} - 48 \, B a^{3} + 56 \, A a^{2} b - 3 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{2} + 4 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}}{105 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*B*b^3*x^3 - 48*B*a^3 + 56*A*a^2*b - 3*(6*B*a*b^2 - 7*A*b^3)*x^2 + 4*(6
*B*a^2*b - 7*A*a*b^2)*x)*sqrt(b*x + a)/b^4

_______________________________________________________________________________________

Sympy [A]  time = 20.1068, size = 240, normalized size = 2.58 \[ \begin{cases} - \frac{\frac{2 A a \left (\frac{a^{2}}{\sqrt{a + b x}} + 2 a \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{3}\right )}{b^{2}} + \frac{2 A \left (- \frac{a^{3}}{\sqrt{a + b x}} - 3 a^{2} \sqrt{a + b x} + a \left (a + b x\right )^{\frac{3}{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{2}} + \frac{2 B a \left (- \frac{a^{3}}{\sqrt{a + b x}} - 3 a^{2} \sqrt{a + b x} + a \left (a + b x\right )^{\frac{3}{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{3}} + \frac{2 B \left (\frac{a^{4}}{\sqrt{a + b x}} + 4 a^{3} \sqrt{a + b x} - 2 a^{2} \left (a + b x\right )^{\frac{3}{2}} + \frac{4 a \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{3}}}{b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{4}}{4}}{\sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-(2*A*a*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)
/b**2 + 2*A*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (
a + b*x)**(5/2)/5)/b**2 + 2*B*a*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*
(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**3 + 2*B*(a**4/sqrt(a + b*x) + 4*a**3*s
qrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7/
2)/7)/b**3)/b, Ne(b, 0)), ((A*x**3/3 + B*x**4/4)/sqrt(a), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231039, size = 155, normalized size = 1.67 \[ \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{8} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{8} + 15 \, \sqrt{b x + a} a^{2} b^{8}\right )} A}{b^{10}} + \frac{3 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{18} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{18} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{18} - 35 \, \sqrt{b x + a} a^{3} b^{18}\right )} B}{b^{21}}\right )}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(7*(3*(b*x + a)^(5/2)*b^8 - 10*(b*x + a)^(3/2)*a*b^8 + 15*sqrt(b*x + a)*a^
2*b^8)*A/b^10 + 3*(5*(b*x + a)^(7/2)*b^18 - 21*(b*x + a)^(5/2)*a*b^18 + 35*(b*x
+ a)^(3/2)*a^2*b^18 - 35*sqrt(b*x + a)*a^3*b^18)*B/b^21)/b