3.117 \(\int \frac{A+B x}{x^4 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{16 c^2 \sqrt{b x+c x^2} (7 b B-6 A c)}{105 b^4 x}+\frac{8 c \sqrt{b x+c x^2} (7 b B-6 A c)}{105 b^3 x^2}-\frac{2 \sqrt{b x+c x^2} (7 b B-6 A c)}{35 b^2 x^3}-\frac{2 A \sqrt{b x+c x^2}}{7 b x^4} \]

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(7*b*x^4) - (2*(7*b*B - 6*A*c)*Sqrt[b*x + c*x^2])/(35*b^2*x^3) + (8*c*(7*b*B - 6*A*c)
*Sqrt[b*x + c*x^2])/(105*b^3*x^2) - (16*c^2*(7*b*B - 6*A*c)*Sqrt[b*x + c*x^2])/(105*b^4*x)

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Rubi [A]  time = 0.109451, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 658, 650} \[ -\frac{16 c^2 \sqrt{b x+c x^2} (7 b B-6 A c)}{105 b^4 x}+\frac{8 c \sqrt{b x+c x^2} (7 b B-6 A c)}{105 b^3 x^2}-\frac{2 \sqrt{b x+c x^2} (7 b B-6 A c)}{35 b^2 x^3}-\frac{2 A \sqrt{b x+c x^2}}{7 b x^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^4*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(7*b*x^4) - (2*(7*b*B - 6*A*c)*Sqrt[b*x + c*x^2])/(35*b^2*x^3) + (8*c*(7*b*B - 6*A*c)
*Sqrt[b*x + c*x^2])/(105*b^3*x^2) - (16*c^2*(7*b*B - 6*A*c)*Sqrt[b*x + c*x^2])/(105*b^4*x)

Rule 792

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

Rule 658

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

Rule 650

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

Rubi steps

\begin{align*} \int \frac{A+B x}{x^4 \sqrt{b x+c x^2}} \, dx &=-\frac{2 A \sqrt{b x+c x^2}}{7 b x^4}+\frac{\left (2 \left (-4 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right )\right ) \int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx}{7 b}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{7 b x^4}-\frac{2 (7 b B-6 A c) \sqrt{b x+c x^2}}{35 b^2 x^3}-\frac{(4 c (7 b B-6 A c)) \int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx}{35 b^2}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{7 b x^4}-\frac{2 (7 b B-6 A c) \sqrt{b x+c x^2}}{35 b^2 x^3}+\frac{8 c (7 b B-6 A c) \sqrt{b x+c x^2}}{105 b^3 x^2}+\frac{\left (8 c^2 (7 b B-6 A c)\right ) \int \frac{1}{x \sqrt{b x+c x^2}} \, dx}{105 b^3}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{7 b x^4}-\frac{2 (7 b B-6 A c) \sqrt{b x+c x^2}}{35 b^2 x^3}+\frac{8 c (7 b B-6 A c) \sqrt{b x+c x^2}}{105 b^3 x^2}-\frac{16 c^2 (7 b B-6 A c) \sqrt{b x+c x^2}}{105 b^4 x}\\ \end{align*}

Mathematica [A]  time = 0.0320801, size = 79, normalized size = 0.63 \[ -\frac{2 \sqrt{x (b+c x)} \left (3 A \left (-6 b^2 c x+5 b^3+8 b c^2 x^2-16 c^3 x^3\right )+7 b B x \left (3 b^2-4 b c x+8 c^2 x^2\right )\right )}{105 b^4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^4*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[x*(b + c*x)]*(7*b*B*x*(3*b^2 - 4*b*c*x + 8*c^2*x^2) + 3*A*(5*b^3 - 6*b^2*c*x + 8*b*c^2*x^2 - 16*c^3*x
^3)))/(105*b^4*x^4)

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Maple [A]  time = 0.006, size = 86, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -48\,A{x}^{3}{c}^{3}+56\,B{x}^{3}b{c}^{2}+24\,A{x}^{2}b{c}^{2}-28\,B{x}^{2}{b}^{2}c-18\,A{b}^{2}cx+21\,{b}^{3}Bx+15\,A{b}^{3} \right ) }{105\,{x}^{3}{b}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^4/(c*x^2+b*x)^(1/2),x)

[Out]

-2/105*(c*x+b)*(-48*A*c^3*x^3+56*B*b*c^2*x^3+24*A*b*c^2*x^2-28*B*b^2*c*x^2-18*A*b^2*c*x+21*B*b^3*x+15*A*b^3)/x
^3/b^4/(c*x^2+b*x)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.82936, size = 185, normalized size = 1.48 \begin{align*} -\frac{2 \,{\left (15 \, A b^{3} + 8 \,{\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} - 4 \,{\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}}{105 \, b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

-2/105*(15*A*b^3 + 8*(7*B*b*c^2 - 6*A*c^3)*x^3 - 4*(7*B*b^2*c - 6*A*b*c^2)*x^2 + 3*(7*B*b^3 - 6*A*b^2*c)*x)*sq
rt(c*x^2 + b*x)/(b^4*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{x^{4} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**4/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((A + B*x)/(x**4*sqrt(x*(b + c*x))), x)

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Giac [A]  time = 1.1479, size = 258, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B c + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b \sqrt{c} + 210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A c^{\frac{3}{2}} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{2} + 252 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b c + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{2} \sqrt{c} + 15 \, A b^{3}\right )}}{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

2/105*(140*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*c + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*sqrt(c) + 210*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(3/2) + 21*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^2 + 252*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^2*A*b*c + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*sqrt(c) + 15*A*b^3)/(sqrt(c)*x - sqrt(c*x^2
+ b*x))^7