3.118 \(\int \frac{A+B x}{x^5 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{32 c^3 \sqrt{b x+c x^2} (9 b B-8 A c)}{315 b^5 x}-\frac{16 c^2 \sqrt{b x+c x^2} (9 b B-8 A c)}{315 b^4 x^2}+\frac{4 c \sqrt{b x+c x^2} (9 b B-8 A c)}{105 b^3 x^3}-\frac{2 \sqrt{b x+c x^2} (9 b B-8 A c)}{63 b^2 x^4}-\frac{2 A \sqrt{b x+c x^2}}{9 b x^5} \]

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(9*b*x^5) - (2*(9*b*B - 8*A*c)*Sqrt[b*x + c*x^2])/(63*b^2*x^4) + (4*c*(9*b*B - 8*A*c)
*Sqrt[b*x + c*x^2])/(105*b^3*x^3) - (16*c^2*(9*b*B - 8*A*c)*Sqrt[b*x + c*x^2])/(315*b^4*x^2) + (32*c^3*(9*b*B
- 8*A*c)*Sqrt[b*x + c*x^2])/(315*b^5*x)

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Rubi [A]  time = 0.142561, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 658, 650} \[ \frac{32 c^3 \sqrt{b x+c x^2} (9 b B-8 A c)}{315 b^5 x}-\frac{16 c^2 \sqrt{b x+c x^2} (9 b B-8 A c)}{315 b^4 x^2}+\frac{4 c \sqrt{b x+c x^2} (9 b B-8 A c)}{105 b^3 x^3}-\frac{2 \sqrt{b x+c x^2} (9 b B-8 A c)}{63 b^2 x^4}-\frac{2 A \sqrt{b x+c x^2}}{9 b x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(9*b*x^5) - (2*(9*b*B - 8*A*c)*Sqrt[b*x + c*x^2])/(63*b^2*x^4) + (4*c*(9*b*B - 8*A*c)
*Sqrt[b*x + c*x^2])/(105*b^3*x^3) - (16*c^2*(9*b*B - 8*A*c)*Sqrt[b*x + c*x^2])/(315*b^4*x^2) + (32*c^3*(9*b*B
- 8*A*c)*Sqrt[b*x + c*x^2])/(315*b^5*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^5 \sqrt{b x+c x^2}} \, dx &=-\frac{2 A \sqrt{b x+c x^2}}{9 b x^5}+\frac{\left (2 \left (-5 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right )\right ) \int \frac{1}{x^4 \sqrt{b x+c x^2}} \, dx}{9 b}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{9 b x^5}-\frac{2 (9 b B-8 A c) \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{(2 c (9 b B-8 A c)) \int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx}{21 b^2}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{9 b x^5}-\frac{2 (9 b B-8 A c) \sqrt{b x+c x^2}}{63 b^2 x^4}+\frac{4 c (9 b B-8 A c) \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{\left (8 c^2 (9 b B-8 A c)\right ) \int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx}{105 b^3}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{9 b x^5}-\frac{2 (9 b B-8 A c) \sqrt{b x+c x^2}}{63 b^2 x^4}+\frac{4 c (9 b B-8 A c) \sqrt{b x+c x^2}}{105 b^3 x^3}-\frac{16 c^2 (9 b B-8 A c) \sqrt{b x+c x^2}}{315 b^4 x^2}-\frac{\left (16 c^3 (9 b B-8 A c)\right ) \int \frac{1}{x \sqrt{b x+c x^2}} \, dx}{315 b^4}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{9 b x^5}-\frac{2 (9 b B-8 A c) \sqrt{b x+c x^2}}{63 b^2 x^4}+\frac{4 c (9 b B-8 A c) \sqrt{b x+c x^2}}{105 b^3 x^3}-\frac{16 c^2 (9 b B-8 A c) \sqrt{b x+c x^2}}{315 b^4 x^2}+\frac{32 c^3 (9 b B-8 A c) \sqrt{b x+c x^2}}{315 b^5 x}\\ \end{align*}

Mathematica [A]  time = 0.0458528, size = 100, normalized size = 0.62 \[ -\frac{2 \sqrt{x (b+c x)} \left (A \left (48 b^2 c^2 x^2-40 b^3 c x+35 b^4-64 b c^3 x^3+128 c^4 x^4\right )+9 b B x \left (-6 b^2 c x+5 b^3+8 b c^2 x^2-16 c^3 x^3\right )\right )}{315 b^5 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x*(b + c*x)]*(9*b*B*x*(5*b^3 - 6*b^2*c*x + 8*b*c^2*x^2 - 16*c^3*x^3) + A*(35*b^4 - 40*b^3*c*x + 48*b^
2*c^2*x^2 - 64*b*c^3*x^3 + 128*c^4*x^4)))/(315*b^5*x^5)

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Maple [A]  time = 0.006, size = 110, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,A{c}^{4}{x}^{4}-144\,Bb{c}^{3}{x}^{4}-64\,Ab{c}^{3}{x}^{3}+72\,B{b}^{2}{c}^{2}{x}^{3}+48\,A{b}^{2}{c}^{2}{x}^{2}-54\,B{b}^{3}c{x}^{2}-40\,A{b}^{3}cx+45\,{b}^{4}Bx+35\,A{b}^{4} \right ) }{315\,{x}^{4}{b}^{5}}{\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^5/(c*x^2+b*x)^(1/2),x)

[Out]

-2/315*(c*x+b)*(128*A*c^4*x^4-144*B*b*c^3*x^4-64*A*b*c^3*x^3+72*B*b^2*c^2*x^3+48*A*b^2*c^2*x^2-54*B*b^3*c*x^2-
40*A*b^3*c*x+45*B*b^4*x+35*A*b^4)/x^4/b^5/(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^5/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.87191, size = 235, normalized size = 1.47 \begin{align*} -\frac{2 \,{\left (35 \, A b^{4} - 16 \,{\left (9 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} + 8 \,{\left (9 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 6 \,{\left (9 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} + 5 \,{\left (9 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x}}{315 \, b^{5} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(35*A*b^4 - 16*(9*B*b*c^3 - 8*A*c^4)*x^4 + 8*(9*B*b^2*c^2 - 8*A*b*c^3)*x^3 - 6*(9*B*b^3*c - 8*A*b^2*c^2
)*x^2 + 5*(9*B*b^4 - 8*A*b^3*c)*x)*sqrt(c*x^2 + b*x)/(b^5*x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{x^{5} \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**5/(c*x**2+b*x)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.13886, size = 339, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B c^{\frac{3}{2}} + 756 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b c + 1008 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{2} \sqrt{c} + 1680 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b c^{\frac{3}{2}} + 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{3} + 1080 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{2} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{3} \sqrt{c} + 35 \, A b^{4}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

2/315*(630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^(3/2) + 756*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c + 1008*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^4*A*c^2 + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*sqrt(c) + 1680*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*A*b*c^(3/2) + 45*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3 + 1080*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*A*b^2*c + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*sqrt(c) + 35*A*b^4)/(sqrt(c)*x - sqrt(c*x^2 + b
*x))^9