3.2.56 \(\int \frac {(A+B x) (b x+c x^2)^2}{x^{7/2}} \, dx\)

Optimal. Leaf size=59 \[ -\frac {2 A b^2}{\sqrt {x}}+\frac {2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt {x} (2 A c+b B)+\frac {2}{5} B c^2 x^{5/2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 A b^2}{\sqrt {x}}+\frac {2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt {x} (2 A c+b B)+\frac {2}{5} B c^2 x^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^(7/2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 54, normalized size = 0.92 \begin {gather*} \frac {10 A \left (-3 b^2+6 b c x+c^2 x^2\right )+2 B x \left (15 b^2+10 b c x+3 c^2 x^2\right )}{15 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^(7/2),x]

[Out]

(10*A*(-3*b^2 + 6*b*c*x + c^2*x^2) + 2*B*x*(15*b^2 + 10*b*c*x + 3*c^2*x^2))/(15*Sqrt[x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.05, size = 55, normalized size = 0.93 \begin {gather*} \frac {2 \left (-15 A b^2+30 A b c x+5 A c^2 x^2+15 b^2 B x+10 b B c x^2+3 B c^2 x^3\right )}{15 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^2)/x^(7/2),x]

[Out]

(2*(-15*A*b^2 + 15*b^2*B*x + 30*A*b*c*x + 10*b*B*c*x^2 + 5*A*c^2*x^2 + 3*B*c^2*x^3))/(15*Sqrt[x])

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 51, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} x^{3} - 15 \, A b^{2} + 5 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \, {\left (B b^{2} + 2 \, A b c\right )} x\right )}}{15 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(7/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 53, normalized size = 0.90 \begin {gather*} \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} + \frac {4}{3} \, B b c x^{\frac {3}{2}} + \frac {2}{3} \, A c^{2} x^{\frac {3}{2}} + 2 \, B b^{2} \sqrt {x} + 4 \, A b c \sqrt {x} - \frac {2 \, A b^{2}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(7/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 52, normalized size = 0.88 \begin {gather*} -\frac {2 \left (-3 B \,c^{2} x^{3}-5 A \,c^{2} x^{2}-10 B b c \,x^{2}-30 A b c x -15 B \,b^{2} x +15 A \,b^{2}\right )}{15 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^(7/2),x)

[Out]

-2/15/x^(1/2)*(-3*B*c^2*x^3-5*A*c^2*x^2-10*B*b*c*x^2-30*A*b*c*x-15*B*b^2*x+15*A*b^2)

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 51, normalized size = 0.86 \begin {gather*} \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} - \frac {2 \, A b^{2}}{\sqrt {x}} + \frac {2}{3} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(7/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 51, normalized size = 0.86 \begin {gather*} \sqrt {x}\,\left (2\,B\,b^2+4\,A\,c\,b\right )+x^{3/2}\,\left (\frac {2\,A\,c^2}{3}+\frac {4\,B\,b\,c}{3}\right )-\frac {2\,A\,b^2}{\sqrt {x}}+\frac {2\,B\,c^2\,x^{5/2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^2*(A + B*x))/x^(7/2),x)

[Out]

x^(1/2)*(2*B*b^2 + 4*A*b*c) + x^(3/2)*((2*A*c^2)/3 + (4*B*b*c)/3) - (2*A*b^2)/x^(1/2) + (2*B*c^2*x^(5/2))/5

________________________________________________________________________________________

sympy [A]  time = 2.90, size = 75, normalized size = 1.27 \begin {gather*} - \frac {2 A b^{2}}{\sqrt {x}} + 4 A b c \sqrt {x} + \frac {2 A c^{2} x^{\frac {3}{2}}}{3} + 2 B b^{2} \sqrt {x} + \frac {4 B b c x^{\frac {3}{2}}}{3} + \frac {2 B c^{2} x^{\frac {5}{2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/x**(7/2),x)

[Out]

-2*A*b**2/sqrt(x) + 4*A*b*c*sqrt(x) + 2*A*c**2*x**(3/2)/3 + 2*B*b**2*sqrt(x) + 4*B*b*c*x**(3/2)/3 + 2*B*c**2*x
**(5/2)/5

________________________________________________________________________________________