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

Optimal. Leaf size=83 \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{9} b^3 B x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.0373372, antiderivative size = 83, 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} \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{9} b^3 B x^{9/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0246229, size = 70, normalized size = 0.84 \[ \frac{2}{315} \sqrt{x} \left (63 a^2 b x (5 A+3 B x)+105 a^3 (3 A+B x)+27 a b^2 x^2 (7 A+5 B x)+5 b^3 x^3 (9 A+7 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(105*a^3*(3*A + B*x) + 63*a^2*b*x*(5*A + 3*B*x) + 27*a*b^2*x^2*(7*A + 5*B*x) + 5*b^3*x^3*(9*A + 7*B
*x)))/315

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Maple [A]  time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{70\,B{b}^{3}{x}^{4}+90\,A{b}^{3}{x}^{3}+270\,B{x}^{3}a{b}^{2}+378\,aA{b}^{2}{x}^{2}+378\,B{x}^{2}{a}^{2}b+630\,{a}^{2}Abx+210\,{a}^{3}Bx+630\,{a}^{3}A}{315}\sqrt{x}} \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]

2/315*x^(1/2)*(35*B*b^3*x^4+45*A*b^3*x^3+135*B*a*b^2*x^3+189*A*a*b^2*x^2+189*B*a^2*b*x^2+315*A*a^2*b*x+105*B*a
^3*x+315*A*a^3)

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Maxima [A]  time = 1.09674, size = 99, normalized size = 1.19 \begin{align*} \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + 2 \, A a^{3} \sqrt{x} + \frac{2}{7} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} 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="maxima")

[Out]

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

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Fricas [A]  time = 2.27004, size = 174, normalized size = 2.1 \begin{align*} \frac{2}{315} \,{\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\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]

2/315*(35*B*b^3*x^4 + 315*A*a^3 + 45*(3*B*a*b^2 + A*b^3)*x^3 + 189*(B*a^2*b + A*a*b^2)*x^2 + 105*(B*a^3 + 3*A*
a^2*b)*x)*sqrt(x)

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Sympy [A]  time = 1.18233, size = 110, normalized size = 1.33 \begin{align*} 2 A a^{3} \sqrt{x} + 2 A a^{2} b x^{\frac{3}{2}} + \frac{6 A a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 A b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B a^{3} x^{\frac{3}{2}}}{3} + \frac{6 B a^{2} b x^{\frac{5}{2}}}{5} + \frac{6 B a b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{9}{2}}}{9} \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*sqrt(x) + 2*A*a**2*b*x**(3/2) + 6*A*a*b**2*x**(5/2)/5 + 2*A*b**3*x**(7/2)/7 + 2*B*a**3*x**(3/2)/3 + 6
*B*a**2*b*x**(5/2)/5 + 6*B*a*b**2*x**(7/2)/7 + 2*B*b**3*x**(9/2)/9

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Giac [A]  time = 1.15491, size = 104, normalized size = 1.25 \begin{align*} \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B a b^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B a^{2} b x^{\frac{5}{2}} + \frac{6}{5} \, A a b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{2} b x^{\frac{3}{2}} + 2 \, A a^{3} \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="giac")

[Out]

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