3.106 \(\int \frac{(A+B x) (b x+c x^2)^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{9009 b^4 x^7}+\frac{8 c \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{1287 b^3 x^8}-\frac{2 \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{143 b^2 x^9}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}} \]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(13*b*x^10) - (2*(13*b*B - 6*A*c)*(b*x + c*x^2)^(7/2))/(143*b^2*x^9) + (8*c*(13*b*B
- 6*A*c)*(b*x + c*x^2)^(7/2))/(1287*b^3*x^8) - (16*c^2*(13*b*B - 6*A*c)*(b*x + c*x^2)^(7/2))/(9009*b^4*x^7)

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Rubi [A]  time = 0.121892, 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} \[ -\frac{16 c^2 \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{9009 b^4 x^7}+\frac{8 c \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{1287 b^3 x^8}-\frac{2 \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{143 b^2 x^9}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^10,x]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(13*b*x^10) - (2*(13*b*B - 6*A*c)*(b*x + c*x^2)^(7/2))/(143*b^2*x^9) + (8*c*(13*b*B
- 6*A*c)*(b*x + c*x^2)^(7/2))/(1287*b^3*x^8) - (16*c^2*(13*b*B - 6*A*c)*(b*x + c*x^2)^(7/2))/(9009*b^4*x^7)

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) \left (b x+c x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}+\frac{\left (2 \left (-10 (-b B+A c)+\frac{7}{2} (-b B+2 A c)\right )\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^9} \, dx}{13 b}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}-\frac{2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{143 b^2 x^9}-\frac{(4 c (13 b B-6 A c)) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^8} \, dx}{143 b^2}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}-\frac{2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{143 b^2 x^9}+\frac{8 c (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{1287 b^3 x^8}+\frac{\left (8 c^2 (13 b B-6 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{1287 b^3}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}-\frac{2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{143 b^2 x^9}+\frac{8 c (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{1287 b^3 x^8}-\frac{16 c^2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{9009 b^4 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0316817, size = 86, normalized size = 0.69 \[ -\frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (3 A \left (-126 b^2 c x+231 b^3+56 b c^2 x^2-16 c^3 x^3\right )+13 b B x \left (63 b^2-28 b c x+8 c^2 x^2\right )\right )}{9009 b^4 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^10,x]

[Out]

(-2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(13*b*B*x*(63*b^2 - 28*b*c*x + 8*c^2*x^2) + 3*A*(231*b^3 - 126*b^2*c*x + 56*
b*c^2*x^2 - 16*c^3*x^3)))/(9009*b^4*x^7)

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Maple [A]  time = 0.004, size = 86, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -48\,A{x}^{3}{c}^{3}+104\,B{x}^{3}b{c}^{2}+168\,A{x}^{2}b{c}^{2}-364\,B{x}^{2}{b}^{2}c-378\,A{b}^{2}cx+819\,{b}^{3}Bx+693\,A{b}^{3} \right ) }{9009\,{x}^{9}{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(5/2)/x^10,x)

[Out]

-2/9009*(c*x+b)*(-48*A*c^3*x^3+104*B*b*c^2*x^3+168*A*b*c^2*x^2-364*B*b^2*c*x^2-378*A*b^2*c*x+819*B*b^3*x+693*A
*b^3)*(c*x^2+b*x)^(5/2)/x^9/b^4

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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)^(5/2)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88053, size = 350, normalized size = 2.8 \begin{align*} -\frac{2 \,{\left (693 \, A b^{6} + 8 \,{\left (13 \, B b c^{5} - 6 \, A c^{6}\right )} x^{6} - 4 \,{\left (13 \, B b^{2} c^{4} - 6 \, A b c^{5}\right )} x^{5} + 3 \,{\left (13 \, B b^{3} c^{3} - 6 \, A b^{2} c^{4}\right )} x^{4} +{\left (1469 \, B b^{4} c^{2} + 15 \, A b^{3} c^{3}\right )} x^{3} + 7 \,{\left (299 \, B b^{5} c + 159 \, A b^{4} c^{2}\right )} x^{2} + 63 \,{\left (13 \, B b^{6} + 27 \, A b^{5} c\right )} x\right )} \sqrt{c x^{2} + b x}}{9009 \, b^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^10,x, algorithm="fricas")

[Out]

-2/9009*(693*A*b^6 + 8*(13*B*b*c^5 - 6*A*c^6)*x^6 - 4*(13*B*b^2*c^4 - 6*A*b*c^5)*x^5 + 3*(13*B*b^3*c^3 - 6*A*b
^2*c^4)*x^4 + (1469*B*b^4*c^2 + 15*A*b^3*c^3)*x^3 + 7*(299*B*b^5*c + 159*A*b^4*c^2)*x^2 + 63*(13*B*b^6 + 27*A*
b^5*c)*x)*sqrt(c*x^2 + b*x)/(b^4*x^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{10}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**10,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**10, x)

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Giac [B]  time = 1.16835, size = 744, normalized size = 5.95 \begin{align*} \frac{2 \,{\left (12012 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{10} B c^{4} + 63063 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} B b c^{\frac{7}{2}} + 18018 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} A c^{\frac{9}{2}} + 153153 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B b^{2} c^{3} + 108108 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} A b c^{4} + 219219 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b^{3} c^{\frac{5}{2}} + 297297 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A b^{2} c^{\frac{7}{2}} + 199485 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{4} c^{2} + 485199 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b^{3} c^{3} + 117117 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{5} c^{\frac{3}{2}} + 513513 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{4} c^{\frac{5}{2}} + 43043 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{6} c + 363363 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{5} c^{2} + 9009 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{7} \sqrt{c} + 171171 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{6} c^{\frac{3}{2}} + 819 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{8} + 51597 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{7} c + 9009 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{8} \sqrt{c} + 693 \, A b^{9}\right )}}{9009 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^10,x, algorithm="giac")

[Out]

2/9009*(12012*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*c^4 + 63063*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b*c^(7/2) +
18018*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*c^(9/2) + 153153*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^2*c^3 + 1081
08*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b*c^4 + 219219*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^3*c^(5/2) + 297297
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^2*c^(7/2) + 199485*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^4*c^2 + 485199
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^3*c^3 + 117117*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^(3/2) + 513513
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^4*c^(5/2) + 43043*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^6*c + 363363*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^2 + 9009*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^7*sqrt(c) + 171171*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^3*A*b^6*c^(3/2) + 819*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^8 + 51597*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^2*A*b^7*c + 9009*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^8*sqrt(c) + 693*A*b^9)/(sqrt(c)*x - s
qrt(c*x^2 + b*x))^13