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

Optimal. Leaf size=63 \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{11} B c^2 x^{11/2} \]

[Out]

(2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*c)*x^(7/2))/7 + (2*c*(2*b*B + A*c)*x^(9/2))/9 + (2*B*c^2*x^(11/2))/11

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Rubi [A]  time = 0.0299997, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {765} \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{11} B c^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*c)*x^(7/2))/7 + (2*c*(2*b*B + A*c)*x^(9/2))/9 + (2*B*c^2*x^(11/2))/11

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0165894, size = 55, normalized size = 0.87 \[ \frac{2 x^{5/2} \left (11 A \left (63 b^2+90 b c x+35 c^2 x^2\right )+5 B x \left (99 b^2+154 b c x+63 c^2 x^2\right )\right )}{3465} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(11*A*(63*b^2 + 90*b*c*x + 35*c^2*x^2) + 5*B*x*(99*b^2 + 154*b*c*x + 63*c^2*x^2)))/3465

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Maple [A]  time = 0.006, size = 52, normalized size = 0.8 \begin{align*}{\frac{630\,B{c}^{2}{x}^{3}+770\,A{c}^{2}{x}^{2}+1540\,B{x}^{2}bc+1980\,Abcx+990\,{b}^{2}Bx+1386\,A{b}^{2}}{3465}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*x^(5/2)*(315*B*c^2*x^3+385*A*c^2*x^2+770*B*b*c*x^2+990*A*b*c*x+495*B*b^2*x+693*A*b^2)

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Maxima [A]  time = 1.0969, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*c^2*x^(11/2) + 2/5*A*b^2*x^(5/2) + 2/9*(2*B*b*c + A*c^2)*x^(9/2) + 2/7*(B*b^2 + 2*A*b*c)*x^(7/2)

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Fricas [A]  time = 1.75695, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c^{2} x^{5} + 693 \, A b^{2} x^{2} + 385 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \,{\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*B*c^2*x^5 + 693*A*b^2*x^2 + 385*(2*B*b*c + A*c^2)*x^4 + 495*(B*b^2 + 2*A*b*c)*x^3)*sqrt(x)

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Sympy [A]  time = 1.47915, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{4 A b c x^{\frac{7}{2}}}{7} + \frac{2 A c^{2} x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} + \frac{4 B b c x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**2*x**(5/2)/5 + 4*A*b*c*x**(7/2)/7 + 2*A*c**2*x**(9/2)/9 + 2*B*b**2*x**(7/2)/7 + 4*B*b*c*x**(9/2)/9 + 2*
B*c**2*x**(11/2)/11

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Giac [A]  time = 1.12664, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B b c x^{\frac{9}{2}} + \frac{2}{9} \, A c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A b c x^{\frac{7}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*c^2*x^(11/2) + 4/9*B*b*c*x^(9/2) + 2/9*A*c^2*x^(9/2) + 2/7*B*b^2*x^(7/2) + 4/7*A*b*c*x^(7/2) + 2/5*A*b^
2*x^(5/2)