3.338 \(\int \sqrt{x} (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{9} b^2 x^{9/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{11} b^3 B x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.0370037, antiderivative size = 85, 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, Rules used = {76} \[ \frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{3} a^3 A x^{3/2}+\frac{2}{9} b^2 x^{9/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{11} b^3 B x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x)^3*(A + B*x),x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \sqrt{x} (a+b x)^3 (A+B x) \, dx &=\int \left (a^3 A \sqrt{x}+a^2 (3 A b+a B) x^{3/2}+3 a b (A b+a B) x^{5/2}+b^2 (A b+3 a B) x^{7/2}+b^3 B x^{9/2}\right ) \, dx\\ &=\frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^2 (3 A b+a B) x^{5/2}+\frac{6}{7} a b (A b+a B) x^{7/2}+\frac{2}{9} b^2 (A b+3 a B) x^{9/2}+\frac{2}{11} b^3 B x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0231475, size = 71, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (297 a^2 b x (7 A+5 B x)+231 a^3 (5 A+3 B x)+165 a b^2 x^2 (9 A+7 B x)+35 b^3 x^3 (11 A+9 B x)\right )}{3465} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x)^3*(A + B*x),x]

[Out]

(2*x^(3/2)*(231*a^3*(5*A + 3*B*x) + 297*a^2*b*x*(7*A + 5*B*x) + 165*a*b^2*x^2*(9*A + 7*B*x) + 35*b^3*x^3*(11*A
+ 9*B*x)))/3465

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Maple [A]  time = 0.004, size = 76, normalized size = 0.9 \begin{align*}{\frac{630\,B{b}^{3}{x}^{4}+770\,A{b}^{3}{x}^{3}+2310\,B{x}^{3}a{b}^{2}+2970\,aA{b}^{2}{x}^{2}+2970\,B{x}^{2}{a}^{2}b+4158\,{a}^{2}Abx+1386\,{a}^{3}Bx+2310\,{a}^{3}A}{3465}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)*x^(1/2),x)

[Out]

2/3465*x^(3/2)*(315*B*b^3*x^4+385*A*b^3*x^3+1155*B*a*b^2*x^3+1485*A*a*b^2*x^2+1485*B*a^2*b*x^2+2079*A*a^2*b*x+
693*B*a^3*x+1155*A*a^3)

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Maxima [A]  time = 1.08665, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} + \frac{2}{9} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*x^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.26565, size = 186, normalized size = 2.19 \begin{align*} \frac{2}{3465} \,{\left (315 \, B b^{3} x^{5} + 1155 \, A a^{3} x + 385 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 1485 \,{\left (B a^{2} b + A a b^{2}\right )} x^{3} + 693 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*x^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^3*x^5 + 1155*A*a^3*x + 385*(3*B*a*b^2 + A*b^3)*x^4 + 1485*(B*a^2*b + A*a*b^2)*x^3 + 693*(B*a^3
+ 3*A*a^2*b)*x^2)*sqrt(x)

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Sympy [A]  time = 3.78006, size = 95, normalized size = 1.12 \begin{align*} \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 B b^{3} x^{\frac{11}{2}}}{11} + \frac{2 x^{\frac{9}{2}} \left (A b^{3} + 3 B a b^{2}\right )}{9} + \frac{2 x^{\frac{7}{2}} \left (3 A a b^{2} + 3 B a^{2} b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (3 A a^{2} b + B a^{3}\right )}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)*x**(1/2),x)

[Out]

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

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Giac [A]  time = 1.18703, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B a b^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B a^{2} b x^{\frac{7}{2}} + \frac{6}{7} \, A a b^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{6}{5} \, A a^{2} b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*x^(1/2),x, algorithm="giac")

[Out]

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