3.2.8 \(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=195 \[ -\frac {256 c^4 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{765765 b^6 x^7}+\frac {128 c^3 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{109395 b^5 x^8}-\frac {32 c^2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{12155 b^4 x^9}+\frac {16 c \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{3315 b^3 x^{10}}-\frac {2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{255 b^2 x^{11}}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}} \]

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Rubi [A]  time = 0.21, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 658, 650} \begin {gather*} -\frac {256 c^4 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{765765 b^6 x^7}+\frac {128 c^3 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{109395 b^5 x^8}-\frac {32 c^2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{12155 b^4 x^9}+\frac {16 c \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{3315 b^3 x^{10}}-\frac {2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{255 b^2 x^{11}}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^12,x]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(17*b*x^12) - (2*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(255*b^2*x^11) + (16*c*(17*
b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(3315*b^3*x^10) - (32*c^2*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(12155*b^4
*x^9) + (128*c^3*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(109395*b^5*x^8) - (256*c^4*(17*b*B - 10*A*c)*(b*x + c
*x^2)^(7/2))/(765765*b^6*x^7)

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]

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 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]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}+\frac {\left (2 \left (-12 (-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^{11}} \, dx}{17 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}-\frac {2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{255 b^2 x^{11}}-\frac {(8 c (17 b B-10 A c)) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{10}} \, dx}{255 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}-\frac {2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{255 b^2 x^{11}}+\frac {16 c (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{3315 b^3 x^{10}}+\frac {\left (16 c^2 (17 b B-10 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^9} \, dx}{1105 b^3}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}-\frac {2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{255 b^2 x^{11}}+\frac {16 c (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{3315 b^3 x^{10}}-\frac {32 c^2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{12155 b^4 x^9}-\frac {\left (64 c^3 (17 b B-10 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^8} \, dx}{12155 b^4}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}-\frac {2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{255 b^2 x^{11}}+\frac {16 c (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{3315 b^3 x^{10}}-\frac {32 c^2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{12155 b^4 x^9}+\frac {128 c^3 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{109395 b^5 x^8}+\frac {\left (128 c^4 (17 b B-10 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{109395 b^5}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}}-\frac {2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{255 b^2 x^{11}}+\frac {16 c (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{3315 b^3 x^{10}}-\frac {32 c^2 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{12155 b^4 x^9}+\frac {128 c^3 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{109395 b^5 x^8}-\frac {256 c^4 (17 b B-10 A c) \left (b x+c x^2\right )^{7/2}}{765765 b^6 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 130, normalized size = 0.67 \begin {gather*} -\frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (5 A \left (9009 b^5-6006 b^4 c x+3696 b^3 c^2 x^2-2016 b^2 c^3 x^3+896 b c^4 x^4-256 c^5 x^5\right )+17 b B x \left (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\right )\right )}{765765 b^6 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^12,x]

[Out]

(-2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(17*b*B*x*(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) + 5*A*(9009*b^5 - 6006*b^4*c*x + 3696*b^3*c^2*x^2 - 2016*b^2*c^3*x^3 + 896*b*c^4*x^4 - 256*c^5*x^5)))
/(765765*b^6*x^9)

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IntegrateAlgebraic [A]  time = 0.57, size = 204, normalized size = 1.05 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-45045 A b^8-105105 A b^7 c x-63525 A b^6 c^2 x^2-315 A b^5 c^3 x^3+350 A b^4 c^4 x^4-400 A b^3 c^5 x^5+480 A b^2 c^6 x^6-640 A b c^7 x^7+1280 A c^8 x^8-51051 b^8 B x-121737 b^7 B c x^2-76041 b^6 B c^2 x^3-595 b^5 B c^3 x^4+680 b^4 B c^4 x^5-816 b^3 B c^5 x^6+1088 b^2 B c^6 x^7-2176 b B c^7 x^8\right )}{765765 b^6 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^(5/2))/x^12,x]

[Out]

(2*Sqrt[b*x + c*x^2]*(-45045*A*b^8 - 51051*b^8*B*x - 105105*A*b^7*c*x - 121737*b^7*B*c*x^2 - 63525*A*b^6*c^2*x
^2 - 76041*b^6*B*c^2*x^3 - 315*A*b^5*c^3*x^3 - 595*b^5*B*c^3*x^4 + 350*A*b^4*c^4*x^4 + 680*b^4*B*c^4*x^5 - 400
*A*b^3*c^5*x^5 - 816*b^3*B*c^5*x^6 + 480*A*b^2*c^6*x^6 + 1088*b^2*B*c^6*x^7 - 640*A*b*c^7*x^7 - 2176*b*B*c^7*x
^8 + 1280*A*c^8*x^8))/(765765*b^6*x^9)

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fricas [A]  time = 0.41, size = 202, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (45045 \, A b^{8} + 128 \, {\left (17 \, B b c^{7} - 10 \, A c^{8}\right )} x^{8} - 64 \, {\left (17 \, B b^{2} c^{6} - 10 \, A b c^{7}\right )} x^{7} + 48 \, {\left (17 \, B b^{3} c^{5} - 10 \, A b^{2} c^{6}\right )} x^{6} - 40 \, {\left (17 \, B b^{4} c^{4} - 10 \, A b^{3} c^{5}\right )} x^{5} + 35 \, {\left (17 \, B b^{5} c^{3} - 10 \, A b^{4} c^{4}\right )} x^{4} + 63 \, {\left (1207 \, B b^{6} c^{2} + 5 \, A b^{5} c^{3}\right )} x^{3} + 231 \, {\left (527 \, B b^{7} c + 275 \, A b^{6} c^{2}\right )} x^{2} + 3003 \, {\left (17 \, B b^{8} + 35 \, A b^{7} c\right )} x\right )} \sqrt {c x^{2} + b x}}{765765 \, b^{6} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^12,x, algorithm="fricas")

[Out]

-2/765765*(45045*A*b^8 + 128*(17*B*b*c^7 - 10*A*c^8)*x^8 - 64*(17*B*b^2*c^6 - 10*A*b*c^7)*x^7 + 48*(17*B*b^3*c
^5 - 10*A*b^2*c^6)*x^6 - 40*(17*B*b^4*c^4 - 10*A*b^3*c^5)*x^5 + 35*(17*B*b^5*c^3 - 10*A*b^4*c^4)*x^4 + 63*(120
7*B*b^6*c^2 + 5*A*b^5*c^3)*x^3 + 231*(527*B*b^7*c + 275*A*b^6*c^2)*x^2 + 3003*(17*B*b^8 + 35*A*b^7*c)*x)*sqrt(
c*x^2 + b*x)/(b^6*x^9)

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giac [B]  time = 0.24, size = 671, normalized size = 3.44 \begin {gather*} \frac {2 \, {\left (2450448 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{12} B c^{5} + 16336320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11} B b c^{\frac {9}{2}} + 4084080 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11} A c^{\frac {11}{2}} + 49884120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} B b^{2} c^{4} + 29755440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} A b c^{5} + 91126035 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B b^{3} c^{\frac {7}{2}} + 99549450 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} A b^{2} c^{\frac {9}{2}} + 109674565 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b^{4} c^{3} + 200800600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A b^{3} c^{4} + 90513423 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{5} c^{\frac {5}{2}} + 270315045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b^{4} c^{\frac {7}{2}} + 51723945 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{6} c^{2} + 254303595 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{5} c^{3} + 20165145 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{7} c^{\frac {3}{2}} + 170255085 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{6} c^{\frac {5}{2}} + 5124735 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{8} c + 80994375 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{7} c^{2} + 765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{9} \sqrt {c} + 26801775 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{8} c^{\frac {3}{2}} + 51051 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{10} + 5870865 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{9} c + 765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{10} \sqrt {c} + 45045 \, A b^{11}\right )}}{765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^12,x, algorithm="giac")

[Out]

2/765765*(2450448*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*B*c^5 + 16336320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b*c
^(9/2) + 4084080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*c^(11/2) + 49884120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B
*b^2*c^4 + 29755440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b*c^5 + 91126035*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*
b^3*c^(7/2) + 99549450*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^2*c^(9/2) + 109674565*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^8*B*b^4*c^3 + 200800600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^4 + 90513423*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^7*B*b^5*c^(5/2) + 270315045*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^4*c^(7/2) + 51723945*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^6*B*b^6*c^2 + 254303595*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^5*c^3 + 20165145*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^5*B*b^7*c^(3/2) + 170255085*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^6*c^(5/2) + 5124735*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^4*B*b^8*c + 80994375*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^7*c^2 + 765765*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^3*B*b^9*sqrt(c) + 26801775*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^8*c^(3/2) + 51051*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^2*B*b^10 + 5870865*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^9*c + 765765*(sqrt(c)*x - sq
rt(c*x^2 + b*x))*A*b^10*sqrt(c) + 45045*A*b^11)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^17

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maple [A]  time = 0.05, size = 134, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+2176 B b \,c^{4} x^{5}+4480 A b \,c^{4} x^{4}-7616 B \,b^{2} c^{3} x^{4}-10080 A \,b^{2} c^{3} x^{3}+17136 B \,b^{3} c^{2} x^{3}+18480 A \,b^{3} c^{2} x^{2}-31416 B \,b^{4} c \,x^{2}-30030 A \,b^{4} c x +51051 B \,b^{5} x +45045 A \,b^{5}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{765765 b^{6} x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(5/2)/x^12,x)

[Out]

-2/765765*(c*x+b)*(-1280*A*c^5*x^5+2176*B*b*c^4*x^5+4480*A*b*c^4*x^4-7616*B*b^2*c^3*x^4-10080*A*b^2*c^3*x^3+17
136*B*b^3*c^2*x^3+18480*A*b^3*c^2*x^2-31416*B*b^4*c*x^2-30030*A*b^4*c*x+51051*B*b^5*x+45045*A*b^5)*(c*x^2+b*x)
^(5/2)/x^11/b^6

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maxima [B]  time = 1.09, size = 442, normalized size = 2.27 \begin {gather*} -\frac {256 \, \sqrt {c x^{2} + b x} B c^{7}}{45045 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{8}}{153153 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{6}}{45045 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{7}}{153153 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{5}}{15015 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{6}}{51051 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c^{4}}{9009 \, b^{2} x^{4}} - \frac {160 \, \sqrt {c x^{2} + b x} A c^{5}}{153153 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{3}}{1287 \, b x^{5}} + \frac {20 \, \sqrt {c x^{2} + b x} A c^{4}}{21879 \, b^{2} x^{5}} + \frac {\sqrt {c x^{2} + b x} B c^{2}}{715 \, x^{6}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{2431 \, b x^{6}} - \frac {\sqrt {c x^{2} + b x} B b c}{780 \, x^{7}} + \frac {\sqrt {c x^{2} + b x} A c^{2}}{1326 \, x^{7}} - \frac {\sqrt {c x^{2} + b x} B b^{2}}{60 \, x^{8}} - \frac {\sqrt {c x^{2} + b x} A b c}{1428 \, x^{8}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{12 \, x^{9}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{2}}{476 \, x^{9}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{5 \, x^{10}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{84 \, x^{10}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{6 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^12,x, algorithm="maxima")

[Out]

-256/45045*sqrt(c*x^2 + b*x)*B*c^7/(b^5*x) + 512/153153*sqrt(c*x^2 + b*x)*A*c^8/(b^6*x) + 128/45045*sqrt(c*x^2
+ b*x)*B*c^6/(b^4*x^2) - 256/153153*sqrt(c*x^2 + b*x)*A*c^7/(b^5*x^2) - 32/15015*sqrt(c*x^2 + b*x)*B*c^5/(b^3
*x^3) + 64/51051*sqrt(c*x^2 + b*x)*A*c^6/(b^4*x^3) + 16/9009*sqrt(c*x^2 + b*x)*B*c^4/(b^2*x^4) - 160/153153*sq
rt(c*x^2 + b*x)*A*c^5/(b^3*x^4) - 2/1287*sqrt(c*x^2 + b*x)*B*c^3/(b*x^5) + 20/21879*sqrt(c*x^2 + b*x)*A*c^4/(b
^2*x^5) + 1/715*sqrt(c*x^2 + b*x)*B*c^2/x^6 - 2/2431*sqrt(c*x^2 + b*x)*A*c^3/(b*x^6) - 1/780*sqrt(c*x^2 + b*x)
*B*b*c/x^7 + 1/1326*sqrt(c*x^2 + b*x)*A*c^2/x^7 - 1/60*sqrt(c*x^2 + b*x)*B*b^2/x^8 - 1/1428*sqrt(c*x^2 + b*x)*
A*b*c/x^8 + 1/12*(c*x^2 + b*x)^(3/2)*B*b/x^9 - 5/476*sqrt(c*x^2 + b*x)*A*b^2/x^9 - 1/5*(c*x^2 + b*x)^(5/2)*B/x
^10 + 5/84*(c*x^2 + b*x)^(3/2)*A*b/x^10 - 1/6*(c*x^2 + b*x)^(5/2)*A/x^11

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mupad [B]  time = 5.40, size = 372, normalized size = 1.91 \begin {gather*} \frac {20\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{21879\,b^2\,x^5}-\frac {110\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{663\,x^7}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{15\,x^8}-\frac {142\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{715\,x^6}-\frac {2\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{2431\,b\,x^6}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{17\,x^9}-\frac {160\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{153153\,b^3\,x^4}+\frac {64\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{51051\,b^4\,x^3}-\frac {256\,A\,c^7\,\sqrt {c\,x^2+b\,x}}{153153\,b^5\,x^2}+\frac {512\,A\,c^8\,\sqrt {c\,x^2+b\,x}}{153153\,b^6\,x}-\frac {2\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{1287\,b\,x^5}+\frac {16\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{9009\,b^2\,x^4}-\frac {32\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{15015\,b^3\,x^3}+\frac {128\,B\,c^6\,\sqrt {c\,x^2+b\,x}}{45045\,b^4\,x^2}-\frac {256\,B\,c^7\,\sqrt {c\,x^2+b\,x}}{45045\,b^5\,x}-\frac {14\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{51\,x^8}-\frac {62\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{195\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^(5/2)*(A + B*x))/x^12,x)

[Out]

(20*A*c^4*(b*x + c*x^2)^(1/2))/(21879*b^2*x^5) - (110*A*c^2*(b*x + c*x^2)^(1/2))/(663*x^7) - (2*B*b^2*(b*x + c
*x^2)^(1/2))/(15*x^8) - (142*B*c^2*(b*x + c*x^2)^(1/2))/(715*x^6) - (2*A*c^3*(b*x + c*x^2)^(1/2))/(2431*b*x^6)
- (2*A*b^2*(b*x + c*x^2)^(1/2))/(17*x^9) - (160*A*c^5*(b*x + c*x^2)^(1/2))/(153153*b^3*x^4) + (64*A*c^6*(b*x
+ c*x^2)^(1/2))/(51051*b^4*x^3) - (256*A*c^7*(b*x + c*x^2)^(1/2))/(153153*b^5*x^2) + (512*A*c^8*(b*x + c*x^2)^
(1/2))/(153153*b^6*x) - (2*B*c^3*(b*x + c*x^2)^(1/2))/(1287*b*x^5) + (16*B*c^4*(b*x + c*x^2)^(1/2))/(9009*b^2*
x^4) - (32*B*c^5*(b*x + c*x^2)^(1/2))/(15015*b^3*x^3) + (128*B*c^6*(b*x + c*x^2)^(1/2))/(45045*b^4*x^2) - (256
*B*c^7*(b*x + c*x^2)^(1/2))/(45045*b^5*x) - (14*A*b*c*(b*x + c*x^2)^(1/2))/(51*x^8) - (62*B*b*c*(b*x + c*x^2)^
(1/2))/(195*x^7)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{12}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**12,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**12, x)

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