3.2.32 \(\int \frac {x^2 (A+B x)}{(b x+c x^2)^{5/2}} \, dx\) [132]

Optimal. Leaf size=67 \[ -\frac {2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (b B+2 A c) x}{3 b^2 c \sqrt {b x+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {802, 650} \begin {gather*} \frac {2 x (2 A c+b B)}{3 b^2 c \sqrt {b x+c x^2}}-\frac {2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x))/(b*x + c*x^2)^(5/2),x]

[Out]

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

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 802

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Dist[e*((m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 35, normalized size = 0.52 \begin {gather*} \frac {2 x^2 (3 A b+b B x+2 A c x)}{3 b^2 (x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x))/(b*x + c*x^2)^(5/2),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(59)=118\).
time = 0.56, size = 218, normalized size = 3.25

method	result	size
trager	\(\frac {2 \left (2 A c x +b B x +3 A b \right ) \sqrt {c \,x^{2}+b x}}{3 b^{2} \left (c x +b \right )^{2}}\)	\(38\)
gosper	\(\frac {2 x^{3} \left (c x +b \right ) \left (2 A c x +b B x +3 A b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\)	\(39\)
default	\(B \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+A \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )\)	\(218\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(B*x+A)/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

B*(-x^2/c/(c*x^2+b*x)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x)^(3/2)-1/2*b/c*(-2/
3*(2*c*x+b)/b^2/(c*x^2+b*x)^(3/2)+16/3*c*(2*c*x+b)/b^4/(c*x^2+b*x)^(1/2)))))+A*(-1/2*x/c/(c*x^2+b*x)^(3/2)-1/4
*b/c*(-1/3/c/(c*x^2+b*x)^(3/2)-1/2*b/c*(-2/3*(2*c*x+b)/b^2/(c*x^2+b*x)^(3/2)+16/3*c*(2*c*x+b)/b^4/(c*x^2+b*x)^
(1/2))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (59) = 118\).
time = 0.27, size = 134, normalized size = 2.00 \begin {gather*} -\frac {B x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, A x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {B b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, A x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {2 \, B x}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {B}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {2 \, A}{3 \, \sqrt {c x^{2} + b x} b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.22, size = 51, normalized size = 0.76 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (3 \, A b + {\left (B b + 2 \, A c\right )} x\right )}}{3 \, {\left (b^{2} c^{2} x^{2} + 2 \, b^{3} c x + b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(B*x+A)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral(x**2*(A + B*x)/(x*(b + c*x))**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (59) = 118\).
time = 0.83, size = 119, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b \sqrt {c} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A c^{\frac {3}{2}} + B b^{2} + 2 \, A b c\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )}^{3} c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.16, size = 37, normalized size = 0.55 \begin {gather*} \frac {2\,\sqrt {c\,x^2+b\,x}\,\left (3\,A\,b+2\,A\,c\,x+B\,b\,x\right )}{3\,b^2\,{\left (b+c\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(A + B*x))/(b*x + c*x^2)^(5/2),x)

[Out]

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

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