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

Optimal. Leaf size=160 \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^6}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(11*b*x^7) - (2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(99*b^2*x^6) + (4*c*(11*b*B -
8*A*c)*(b*x + c*x^2)^(3/2))/(231*b^3*x^5) - (16*c^2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(1155*b^4*x^4) + (3
2*c^3*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(3465*b^5*x^3)

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Rubi [A]  time = 0.157367, 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} \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^6}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(11*b*x^7) - (2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(99*b^2*x^6) + (4*c*(11*b*B -
8*A*c)*(b*x + c*x^2)^(3/2))/(231*b^3*x^5) - (16*c^2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(1155*b^4*x^4) + (3
2*c^3*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(3465*b^5*x^3)

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) \sqrt{b x+c x^2}}{x^7} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac{\left (2 \left (-7 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int \frac{\sqrt{b x+c x^2}}{x^6} \, dx}{11 b}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac{2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac{(2 c (11 b B-8 A c)) \int \frac{\sqrt{b x+c x^2}}{x^5} \, dx}{33 b^2}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac{2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac{4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac{\left (8 c^2 (11 b B-8 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{x^4} \, dx}{231 b^3}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac{2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac{4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac{16 c^2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac{\left (16 c^3 (11 b B-8 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{x^3} \, dx}{1155 b^4}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac{2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac{4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac{16 c^2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}+\frac{32 c^3 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0524444, size = 100, normalized size = 0.62 \[ -\frac{2 (x (b+c x))^{3/2} \left (A \left (240 b^2 c^2 x^2-280 b^3 c x+315 b^4-192 b c^3 x^3+128 c^4 x^4\right )+11 b B x \left (-30 b^2 c x+35 b^3+24 b c^2 x^2-16 c^3 x^3\right )\right )}{3465 b^5 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(x*(b + c*x))^(3/2)*(11*b*B*x*(35*b^3 - 30*b^2*c*x + 24*b*c^2*x^2 - 16*c^3*x^3) + A*(315*b^4 - 280*b^3*c*x
+ 240*b^2*c^2*x^2 - 192*b*c^3*x^3 + 128*c^4*x^4)))/(3465*b^5*x^7)

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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}-176\,Bb{c}^{3}{x}^{4}-192\,Ab{c}^{3}{x}^{3}+264\,B{b}^{2}{c}^{2}{x}^{3}+240\,A{b}^{2}{c}^{2}{x}^{2}-330\,B{b}^{3}c{x}^{2}-280\,A{b}^{3}cx+385\,{b}^{4}Bx+315\,A{b}^{4} \right ) }{3465\,{x}^{6}{b}^{5}}\sqrt{c{x}^{2}+bx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(1/2)/x^7,x)

[Out]

-2/3465*(c*x+b)*(128*A*c^4*x^4-176*B*b*c^3*x^4-192*A*b*c^3*x^3+264*B*b^2*c^2*x^3+240*A*b^2*c^2*x^2-330*B*b^3*c
*x^2-280*A*b^3*c*x+385*B*b^4*x+315*A*b^4)*(c*x^2+b*x)^(1/2)/x^6/b^5

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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)*(c*x^2+b*x)^(1/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88224, size = 292, normalized size = 1.82 \begin{align*} -\frac{2 \,{\left (315 \, A b^{5} - 16 \,{\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{5} + 8 \,{\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{4} - 6 \,{\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{3} + 5 \,{\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{2} + 35 \,{\left (11 \, B b^{5} + A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x}}{3465 \, b^{5} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^7,x, algorithm="fricas")

[Out]

-2/3465*(315*A*b^5 - 16*(11*B*b*c^4 - 8*A*c^5)*x^5 + 8*(11*B*b^2*c^3 - 8*A*b*c^4)*x^4 - 6*(11*B*b^3*c^2 - 8*A*
b^2*c^3)*x^3 + 5*(11*B*b^4*c - 8*A*b^3*c^2)*x^2 + 35*(11*B*b^5 + A*b^4*c)*x)*sqrt(c*x^2 + b*x)/(b^5*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**7,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**7, x)

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Giac [B]  time = 1.22731, size = 501, normalized size = 3.13 \begin{align*} \frac{2 \,{\left (6930 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B c^{\frac{5}{2}} + 19404 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b c^{2} + 11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A c^{3} + 21945 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{2} c^{\frac{3}{2}} + 36960 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b c^{\frac{5}{2}} + 12375 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{3} c + 51480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{2} c^{2} + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{4} \sqrt{c} + 38115 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{3} c^{\frac{3}{2}} + 385 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{5} + 15785 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{4} c + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{5} \sqrt{c} + 315 \, A b^{6}\right )}}{3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^7,x, algorithm="giac")

[Out]

2/3465*(6930*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*c^(5/2) + 19404*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b*c^2 + 1
1088*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*c^3 + 21945*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^(3/2) + 36960*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b*c^(5/2) + 12375*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*c + 51480*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^2 + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*sqrt(c) + 38115*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*A*b^3*c^(3/2) + 385*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5 + 15785*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^2*A*b^4*c + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*sqrt(c) + 315*A*b^6)/(sqrt(c)*x - sqrt(c*
x^2 + b*x))^11