3.167 \(\int \frac{(A+B x) (b x+c x^2)^3}{x^{11/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 b^2 (3 A c+b B)}{\sqrt{x}}-\frac{2 A b^3}{3 x^{3/2}}+\frac{2}{3} c^2 x^{3/2} (A c+3 b B)+6 b c \sqrt{x} (A c+b B)+\frac{2}{5} B c^3 x^{5/2} \]

[Out]

(-2*A*b^3)/(3*x^(3/2)) - (2*b^2*(b*B + 3*A*c))/Sqrt[x] + 6*b*c*(b*B + A*c)*Sqrt[x] + (2*c^2*(3*b*B + A*c)*x^(3
/2))/3 + (2*B*c^3*x^(5/2))/5

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Rubi [A]  time = 0.0416227, antiderivative size = 81, 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} \[ -\frac{2 b^2 (3 A c+b B)}{\sqrt{x}}-\frac{2 A b^3}{3 x^{3/2}}+\frac{2}{3} c^2 x^{3/2} (A c+3 b B)+6 b c \sqrt{x} (A c+b B)+\frac{2}{5} B c^3 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x^(11/2),x]

[Out]

(-2*A*b^3)/(3*x^(3/2)) - (2*b^2*(b*B + 3*A*c))/Sqrt[x] + 6*b*c*(b*B + A*c)*Sqrt[x] + (2*c^2*(3*b*B + A*c)*x^(3
/2))/3 + (2*B*c^3*x^(5/2))/5

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

Mathematica [A]  time = 0.0221872, size = 74, normalized size = 0.91 \[ \frac{6 B x \left (15 b^2 c x-5 b^3+5 b c^2 x^2+c^3 x^3\right )-10 A \left (9 b^2 c x+b^3-9 b c^2 x^2-c^3 x^3\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^(11/2),x]

[Out]

(-10*A*(b^3 + 9*b^2*c*x - 9*b*c^2*x^2 - c^3*x^3) + 6*B*x*(-5*b^3 + 15*b^2*c*x + 5*b*c^2*x^2 + c^3*x^3))/(15*x^
(3/2))

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Maple [A]  time = 0.004, size = 76, normalized size = 0.9 \begin{align*} -{\frac{-6\,B{c}^{3}{x}^{4}-10\,A{x}^{3}{c}^{3}-30\,B{x}^{3}b{c}^{2}-90\,A{x}^{2}b{c}^{2}-90\,B{x}^{2}{b}^{2}c+90\,A{b}^{2}cx+30\,{b}^{3}Bx+10\,A{b}^{3}}{15}{x}^{-{\frac{3}{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^(11/2),x)

[Out]

-2/15/x^(3/2)*(-3*B*c^3*x^4-5*A*c^3*x^3-15*B*b*c^2*x^3-45*A*b*c^2*x^2-45*B*b^2*c*x^2+45*A*b^2*c*x+15*B*b^3*x+5
*A*b^3)

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Maxima [A]  time = 1.17696, size = 99, normalized size = 1.22 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{3}{2}} + 6 \,{\left (B b^{2} c + A b c^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A b^{3} + 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x\right )}}{3 \, x^{\frac{3}{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^(11/2),x, algorithm="maxima")

[Out]

2/5*B*c^3*x^(5/2) + 2/3*(3*B*b*c^2 + A*c^3)*x^(3/2) + 6*(B*b^2*c + A*b*c^2)*sqrt(x) - 2/3*(A*b^3 + 3*(B*b^3 +
3*A*b^2*c)*x)/x^(3/2)

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Fricas [A]  time = 1.87205, size = 165, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (3 \, B c^{3} x^{4} - 5 \, A b^{3} + 5 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} - 15 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x\right )}}{15 \, x^{\frac{3}{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^(11/2),x, algorithm="fricas")

[Out]

2/15*(3*B*c^3*x^4 - 5*A*b^3 + 5*(3*B*b*c^2 + A*c^3)*x^3 + 45*(B*b^2*c + A*b*c^2)*x^2 - 15*(B*b^3 + 3*A*b^2*c)*
x)/x^(3/2)

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Sympy [A]  time = 12.5957, size = 105, normalized size = 1.3 \begin{align*} - \frac{2 A b^{3}}{3 x^{\frac{3}{2}}} - \frac{6 A b^{2} c}{\sqrt{x}} + 6 A b c^{2} \sqrt{x} + \frac{2 A c^{3} x^{\frac{3}{2}}}{3} - \frac{2 B b^{3}}{\sqrt{x}} + 6 B b^{2} c \sqrt{x} + 2 B b c^{2} x^{\frac{3}{2}} + \frac{2 B c^{3} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**(11/2),x)

[Out]

-2*A*b**3/(3*x**(3/2)) - 6*A*b**2*c/sqrt(x) + 6*A*b*c**2*sqrt(x) + 2*A*c**3*x**(3/2)/3 - 2*B*b**3/sqrt(x) + 6*
B*b**2*c*sqrt(x) + 2*B*b*c**2*x**(3/2) + 2*B*c**3*x**(5/2)/5

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Giac [A]  time = 1.14245, size = 101, normalized size = 1.25 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + 2 \, B b c^{2} x^{\frac{3}{2}} + \frac{2}{3} \, A c^{3} x^{\frac{3}{2}} + 6 \, B b^{2} c \sqrt{x} + 6 \, A b c^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B b^{3} x + 9 \, A b^{2} c x + A b^{3}\right )}}{3 \, x^{\frac{3}{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^(11/2),x, algorithm="giac")

[Out]

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