3.161 \(\int \sqrt{x} (A+B x) (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{9} A b^3 x^{9/2}+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{6}{13} b c x^{13/2} (A c+b B)+\frac{2}{17} B c^3 x^{17/2} \]

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (6*b*c*(b*B + A*c)*x^(13/2))/13 + (2*c^2*(3*b*B + A*
c)*x^(15/2))/15 + (2*B*c^3*x^(17/2))/17

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Rubi [A]  time = 0.0420168, antiderivative size = 85, 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} \[ \frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{9} A b^3 x^{9/2}+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{6}{13} b c x^{13/2} (A c+b B)+\frac{2}{17} B c^3 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (6*b*c*(b*B + A*c)*x^(13/2))/13 + (2*c^2*(3*b*B + A*
c)*x^(15/2))/15 + (2*B*c^3*x^(17/2))/17

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 \sqrt{x} (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^{7/2}+b^2 (b B+3 A c) x^{9/2}+3 b c (b B+A c) x^{11/2}+c^2 (3 b B+A c) x^{13/2}+B c^3 x^{15/2}\right ) \, dx\\ &=\frac{2}{9} A b^3 x^{9/2}+\frac{2}{11} b^2 (b B+3 A c) x^{11/2}+\frac{6}{13} b c (b B+A c) x^{13/2}+\frac{2}{15} c^2 (3 b B+A c) x^{15/2}+\frac{2}{17} B c^3 x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0467388, size = 70, normalized size = 0.82 \[ \frac{2 \left (B x^{9/2} (b+c x)^4-\frac{x^{9/2} \left (1755 b^2 c x+715 b^3+1485 b c^2 x^2+429 c^3 x^3\right ) (9 b B-17 A c)}{6435}\right )}{17 c} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(2*(B*x^(9/2)*(b + c*x)^4 - ((9*b*B - 17*A*c)*x^(9/2)*(715*b^3 + 1755*b^2*c*x + 1485*b*c^2*x^2 + 429*c^3*x^3))
/6435))/(17*c)

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Maple [A]  time = 0.004, size = 76, normalized size = 0.9 \begin{align*}{\frac{12870\,B{c}^{3}{x}^{4}+14586\,A{x}^{3}{c}^{3}+43758\,B{x}^{3}b{c}^{2}+50490\,A{x}^{2}b{c}^{2}+50490\,B{x}^{2}{b}^{2}c+59670\,A{b}^{2}cx+19890\,{b}^{3}Bx+24310\,A{b}^{3}}{109395}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/109395*x^(9/2)*(6435*B*c^3*x^4+7293*A*c^3*x^3+21879*B*b*c^2*x^3+25245*A*b*c^2*x^2+25245*B*b^2*c*x^2+29835*A*
b^2*c*x+9945*B*b^3*x+12155*A*b^3)

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Maxima [A]  time = 1.10298, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{2}{15} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^3*x^(17/2) + 2/9*A*b^3*x^(9/2) + 2/15*(3*B*b*c^2 + A*c^3)*x^(15/2) + 6/13*(B*b^2*c + A*b*c^2)*x^(13/2
) + 2/11*(B*b^3 + 3*A*b^2*c)*x^(11/2)

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Fricas [A]  time = 1.85943, size = 198, normalized size = 2.33 \begin{align*} \frac{2}{109395} \,{\left (6435 \, B c^{3} x^{8} + 12155 \, A b^{3} x^{4} + 7293 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 25245 \,{\left (B b^{2} c + A b c^{2}\right )} x^{6} + 9945 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5}\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)^3*x^(1/2),x, algorithm="fricas")

[Out]

2/109395*(6435*B*c^3*x^8 + 12155*A*b^3*x^4 + 7293*(3*B*b*c^2 + A*c^3)*x^7 + 25245*(B*b^2*c + A*b*c^2)*x^6 + 99
45*(B*b^3 + 3*A*b^2*c)*x^5)*sqrt(x)

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Sympy [A]  time = 3.49618, size = 95, normalized size = 1.12 \begin{align*} \frac{2 A b^{3} x^{\frac{9}{2}}}{9} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} + \frac{2 x^{\frac{15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac{2 x^{\frac{13}{2}} \left (3 A b c^{2} + 3 B b^{2} c\right )}{13} + \frac{2 x^{\frac{11}{2}} \left (3 A b^{2} c + B b^{3}\right )}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**3*x**(9/2)/9 + 2*B*c**3*x**(17/2)/17 + 2*x**(15/2)*(A*c**3 + 3*B*b*c**2)/15 + 2*x**(13/2)*(3*A*b*c**2 +
3*B*b**2*c)/13 + 2*x**(11/2)*(3*A*b**2*c + B*b**3)/11

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Giac [A]  time = 1.13212, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B b c^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B b^{2} c x^{\frac{13}{2}} + \frac{6}{13} \, A b c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A b^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^3*x^(17/2) + 2/5*B*b*c^2*x^(15/2) + 2/15*A*c^3*x^(15/2) + 6/13*B*b^2*c*x^(13/2) + 6/13*A*b*c^2*x^(13/
2) + 2/11*B*b^3*x^(11/2) + 6/11*A*b^2*c*x^(11/2) + 2/9*A*b^3*x^(9/2)