3.107 \(\int \frac{(A+B x) (b x+c x^2)^{5/2}}{x^{11}} \, dx\)

Optimal. Leaf size=160 \[ \frac{32 c^3 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{45045 b^5 x^7}-\frac{16 c^2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{6435 b^4 x^8}+\frac{4 c \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{715 b^3 x^9}-\frac{2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{195 b^2 x^{10}}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}} \]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(15*b*x^11) - (2*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(195*b^2*x^10) + (4*c*(15*b*
B - 8*A*c)*(b*x + c*x^2)^(7/2))/(715*b^3*x^9) - (16*c^2*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(6435*b^4*x^8) +
(32*c^3*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(45045*b^5*x^7)

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Rubi [A]  time = 0.163695, 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} \[ \frac{32 c^3 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{45045 b^5 x^7}-\frac{16 c^2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{6435 b^4 x^8}+\frac{4 c \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{715 b^3 x^9}-\frac{2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{195 b^2 x^{10}}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^11,x]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(15*b*x^11) - (2*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(195*b^2*x^10) + (4*c*(15*b*
B - 8*A*c)*(b*x + c*x^2)^(7/2))/(715*b^3*x^9) - (16*c^2*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(6435*b^4*x^8) +
(32*c^3*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(45045*b^5*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^{11}} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}+\frac{\left (2 \left (-11 (-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^{10}} \, dx}{15 b}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac{2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}-\frac{(2 c (15 b B-8 A c)) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^9} \, dx}{65 b^2}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac{2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac{4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}+\frac{\left (8 c^2 (15 b B-8 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^8} \, dx}{715 b^3}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac{2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac{4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}-\frac{16 c^2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{6435 b^4 x^8}-\frac{\left (16 c^3 (15 b B-8 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{6435 b^4}\\ &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac{2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac{4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}-\frac{16 c^2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{6435 b^4 x^8}+\frac{32 c^3 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{45045 b^5 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0341047, size = 107, normalized size = 0.67 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (A \left (-1008 b^2 c^2 x^2+1848 b^3 c x-3003 b^4+448 b c^3 x^3-128 c^4 x^4\right )+15 b B x \left (126 b^2 c x-231 b^3-56 b c^2 x^2+16 c^3 x^3\right )\right )}{45045 b^5 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^11,x]

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(15*b*B*x*(-231*b^3 + 126*b^2*c*x - 56*b*c^2*x^2 + 16*c^3*x^3) + A*(-3003*b^4
+ 1848*b^3*c*x - 1008*b^2*c^2*x^2 + 448*b*c^3*x^3 - 128*c^4*x^4)))/(45045*b^5*x^8)

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Maple [A]  time = 0.007, size = 110, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,A{c}^{4}{x}^{4}-240\,Bb{c}^{3}{x}^{4}-448\,Ab{c}^{3}{x}^{3}+840\,B{b}^{2}{c}^{2}{x}^{3}+1008\,A{b}^{2}{c}^{2}{x}^{2}-1890\,B{b}^{3}c{x}^{2}-1848\,A{b}^{3}cx+3465\,{b}^{4}Bx+3003\,A{b}^{4} \right ) }{45045\,{x}^{10}{b}^{5}} \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^11,x)

[Out]

-2/45045*(c*x+b)*(128*A*c^4*x^4-240*B*b*c^3*x^4-448*A*b*c^3*x^3+840*B*b^2*c^2*x^3+1008*A*b^2*c^2*x^2-1890*B*b^
3*c*x^2-1848*A*b^3*c*x+3465*B*b^4*x+3003*A*b^4)*(c*x^2+b*x)^(5/2)/x^10/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)^(5/2)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85195, size = 404, normalized size = 2.52 \begin{align*} -\frac{2 \,{\left (3003 \, A b^{7} - 16 \,{\left (15 \, B b c^{6} - 8 \, A c^{7}\right )} x^{7} + 8 \,{\left (15 \, B b^{2} c^{5} - 8 \, A b c^{6}\right )} x^{6} - 6 \,{\left (15 \, B b^{3} c^{4} - 8 \, A b^{2} c^{5}\right )} x^{5} + 5 \,{\left (15 \, B b^{4} c^{3} - 8 \, A b^{3} c^{4}\right )} x^{4} + 35 \,{\left (159 \, B b^{5} c^{2} + A b^{4} c^{3}\right )} x^{3} + 63 \,{\left (135 \, B b^{6} c + 71 \, A b^{5} c^{2}\right )} x^{2} + 231 \,{\left (15 \, B b^{7} + 31 \, A b^{6} c\right )} x\right )} \sqrt{c x^{2} + b x}}{45045 \, b^{5} x^{8}} \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^11,x, algorithm="fricas")

[Out]

-2/45045*(3003*A*b^7 - 16*(15*B*b*c^6 - 8*A*c^7)*x^7 + 8*(15*B*b^2*c^5 - 8*A*b*c^6)*x^6 - 6*(15*B*b^3*c^4 - 8*
A*b^2*c^5)*x^5 + 5*(15*B*b^4*c^3 - 8*A*b^3*c^4)*x^4 + 35*(159*B*b^5*c^2 + A*b^4*c^3)*x^3 + 63*(135*B*b^6*c + 7
1*A*b^5*c^2)*x^2 + 231*(15*B*b^7 + 31*A*b^6*c)*x)*sqrt(c*x^2 + b*x)/(b^5*x^8)

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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^{11}}\, 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**11,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**11, x)

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Giac [B]  time = 1.17546, size = 825, normalized size = 5.16 \begin{align*} \frac{2 \,{\left (90090 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11} B c^{\frac{9}{2}} + 540540 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{10} B b c^{4} + 144144 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{10} A c^{5} + 1486485 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} B b^{2} c^{\frac{7}{2}} + 960960 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} A b c^{\frac{9}{2}} + 2425995 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B b^{3} c^{3} + 2934360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} A b^{2} c^{4} + 2567565 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b^{4} c^{\frac{5}{2}} + 5360355 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A b^{3} c^{\frac{7}{2}} + 1816815 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{5} c^{2} + 6451445 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b^{4} c^{3} + 855855 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{6} c^{\frac{3}{2}} + 5324319 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{5} c^{\frac{5}{2}} + 257985 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{7} c + 3042585 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{6} c^{2} + 45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{8} \sqrt{c} + 1186185 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{7} c^{\frac{3}{2}} + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{9} + 301455 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{8} c + 45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{9} \sqrt{c} + 3003 \, A b^{10}\right )}}{45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{15}} \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^11,x, algorithm="giac")

[Out]

2/45045*(90090*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*c^(9/2) + 540540*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b*c^
4 + 144144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*c^5 + 1486485*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^2*c^(7/2)
+ 960960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b*c^(9/2) + 2425995*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^3 +
2934360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^2*c^4 + 2567565*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^(5/2)
+ 5360355*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^3*c^(7/2) + 1816815*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^5*c
^2 + 6451445*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^3 + 855855*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^6*c^(3
/2) + 5324319*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^5*c^(5/2) + 257985*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^7
*c + 3042585*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^6*c^2 + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^8*sqrt(
c) + 1186185*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^7*c^(3/2) + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^9 +
301455*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^8*c + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^9*sqrt(c) + 3003*
A*b^10)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^15