3.1.90 \(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=125 \[ -\frac {16 c^2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{3465 b^4 x^5}+\frac {8 c \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{693 b^3 x^6}-\frac {2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{99 b^2 x^7}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 125, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {16 c^2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{3465 b^4 x^5}+\frac {8 c \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{693 b^3 x^6}-\frac {2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{99 b^2 x^7}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^8,x]

[Out]

(-2*A*(b*x + c*x^2)^(5/2))/(11*b*x^8) - (2*(11*b*B - 6*A*c)*(b*x + c*x^2)^(5/2))/(99*b^2*x^7) + (8*c*(11*b*B -
6*A*c)*(b*x + c*x^2)^(5/2))/(693*b^3*x^6) - (16*c^2*(11*b*B - 6*A*c)*(b*x + c*x^2)^(5/2))/(3465*b^4*x^5)

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 )^{3/2}}{x^8} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}+\frac {\left (2 \left (-8 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^7} \, dx}{11 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}-\frac {(4 c (11 b B-6 A c)) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx}{99 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}+\frac {8 c (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{693 b^3 x^6}+\frac {\left (8 c^2 (11 b B-6 A c)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{693 b^3}\\ &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}+\frac {8 c (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{693 b^3 x^6}-\frac {16 c^2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{3465 b^4 x^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 79, normalized size = 0.63 \begin {gather*} -\frac {2 (x (b+c x))^{5/2} \left (3 A \left (105 b^3-70 b^2 c x+40 b c^2 x^2-16 c^3 x^3\right )+11 b B x \left (35 b^2-20 b c x+8 c^2 x^2\right )\right )}{3465 b^4 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^8,x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(11*b*B*x*(35*b^2 - 20*b*c*x + 8*c^2*x^2) + 3*A*(105*b^3 - 70*b^2*c*x + 40*b*c^2*x^2 -
16*c^3*x^3)))/(3465*b^4*x^8)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.46, size = 132, normalized size = 1.06 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-315 A b^5-420 A b^4 c x-15 A b^3 c^2 x^2+18 A b^2 c^3 x^3-24 A b c^4 x^4+48 A c^5 x^5-385 b^5 B x-550 b^4 B c x^2-33 b^3 B c^2 x^3+44 b^2 B c^3 x^4-88 b B c^4 x^5\right )}{3465 b^4 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^(3/2))/x^8,x]

[Out]

(2*Sqrt[b*x + c*x^2]*(-315*A*b^5 - 385*b^5*B*x - 420*A*b^4*c*x - 550*b^4*B*c*x^2 - 15*A*b^3*c^2*x^2 - 33*b^3*B
*c^2*x^3 + 18*A*b^2*c^3*x^3 + 44*b^2*B*c^3*x^4 - 24*A*b*c^4*x^4 - 88*b*B*c^4*x^5 + 48*A*c^5*x^5))/(3465*b^4*x^
6)

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 130, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (315 \, A b^{5} + 8 \, {\left (11 \, B b c^{4} - 6 \, A c^{5}\right )} x^{5} - 4 \, {\left (11 \, B b^{2} c^{3} - 6 \, A b c^{4}\right )} x^{4} + 3 \, {\left (11 \, B b^{3} c^{2} - 6 \, A b^{2} c^{3}\right )} x^{3} + 5 \, {\left (110 \, B b^{4} c + 3 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} + 12 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{4} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x, algorithm="fricas")

[Out]

-2/3465*(315*A*b^5 + 8*(11*B*b*c^4 - 6*A*c^5)*x^5 - 4*(11*B*b^2*c^3 - 6*A*b*c^4)*x^4 + 3*(11*B*b^3*c^2 - 6*A*b
^2*c^3)*x^3 + 5*(110*B*b^4*c + 3*A*b^3*c^2)*x^2 + 35*(11*B*b^5 + 12*A*b^4*c)*x)*sqrt(c*x^2 + b*x)/(b^4*x^6)

________________________________________________________________________________________

giac [B]  time = 0.40, size = 431, normalized size = 3.45 \begin {gather*} \frac {2 \, {\left (4620 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B c^{3} + 17325 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b c^{\frac {5}{2}} + 6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A c^{\frac {7}{2}} + 28413 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 30492 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b c^{3} + 25410 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac {3}{2}} + 58905 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac {5}{2}} + 12870 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{4} c + 63855 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{5} \sqrt {c} + 41580 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{6} + 16170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{5} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{6} \sqrt {c} + 315 \, A b^{7}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x, algorithm="giac")

[Out]

2/3465*(4620*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^3 + 17325*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b*c^(5/2) + 6
930*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*c^(7/2) + 28413*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^2 + 30492*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^3 + 25410*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^(3/2) + 58905*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^(5/2) + 12870*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^4*c + 63855*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^4*A*b^3*c^2 + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*sqrt(c) + 41580*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^3*A*b^4*c^(3/2) + 385*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6 + 16170*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^2*A*b^5*c + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*sqrt(c) + 315*A*b^7)/(sqrt(c)*x - sqrt(c*x^2
+ b*x))^11

________________________________________________________________________________________

maple [A]  time = 0.05, size = 86, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-48 A \,c^{3} x^{3}+88 B b \,c^{2} x^{3}+120 A b \,c^{2} x^{2}-220 B \,b^{2} c \,x^{2}-210 A \,b^{2} c x +385 B \,b^{3} x +315 A \,b^{3}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3465 b^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x)

[Out]

-2/3465*(c*x+b)*(-48*A*c^3*x^3+88*B*b*c^2*x^3+120*A*b*c^2*x^2-220*B*b^2*c*x^2-210*A*b^2*c*x+385*B*b^3*x+315*A*
b^3)*(c*x^2+b*x)^(3/2)/x^7/b^4

________________________________________________________________________________________

maxima [B]  time = 0.96, size = 268, normalized size = 2.14 \begin {gather*} -\frac {16 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{3} x} + \frac {32 \, \sqrt {c x^{2} + b x} A c^{5}}{1155 \, b^{4} x} + \frac {8 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{2} x^{2}} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{4}}{1155 \, b^{3} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b x^{3}} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{3}}{385 \, b^{2} x^{3}} + \frac {\sqrt {c x^{2} + b x} B c}{63 \, x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{2}}{231 \, b x^{4}} + \frac {\sqrt {c x^{2} + b x} B b}{9 \, x^{5}} + \frac {\sqrt {c x^{2} + b x} A c}{132 \, x^{5}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{3 \, x^{6}} + \frac {3 \, \sqrt {c x^{2} + b x} A b}{44 \, x^{6}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{4 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/x^8,x, algorithm="maxima")

[Out]

-16/315*sqrt(c*x^2 + b*x)*B*c^4/(b^3*x) + 32/1155*sqrt(c*x^2 + b*x)*A*c^5/(b^4*x) + 8/315*sqrt(c*x^2 + b*x)*B*
c^3/(b^2*x^2) - 16/1155*sqrt(c*x^2 + b*x)*A*c^4/(b^3*x^2) - 2/105*sqrt(c*x^2 + b*x)*B*c^2/(b*x^3) + 4/385*sqrt
(c*x^2 + b*x)*A*c^3/(b^2*x^3) + 1/63*sqrt(c*x^2 + b*x)*B*c/x^4 - 2/231*sqrt(c*x^2 + b*x)*A*c^2/(b*x^4) + 1/9*s
qrt(c*x^2 + b*x)*B*b/x^5 + 1/132*sqrt(c*x^2 + b*x)*A*c/x^5 - 1/3*(c*x^2 + b*x)^(3/2)*B/x^6 + 3/44*sqrt(c*x^2 +
b*x)*A*b/x^6 - 1/4*(c*x^2 + b*x)^(3/2)*A/x^7

________________________________________________________________________________________

mupad [B]  time = 2.88, size = 234, normalized size = 1.87 \begin {gather*} \frac {4\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{385\,b^2\,x^3}-\frac {8\,A\,c\,\sqrt {c\,x^2+b\,x}}{33\,x^5}-\frac {2\,B\,b\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {20\,B\,c\,\sqrt {c\,x^2+b\,x}}{63\,x^4}-\frac {2\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{231\,b\,x^4}-\frac {2\,A\,b\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {16\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^2}+\frac {32\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{1155\,b^4\,x}-\frac {2\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b\,x^3}+\frac {8\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{315\,b^2\,x^2}-\frac {16\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{315\,b^3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^(3/2)*(A + B*x))/x^8,x)

[Out]

(4*A*c^3*(b*x + c*x^2)^(1/2))/(385*b^2*x^3) - (8*A*c*(b*x + c*x^2)^(1/2))/(33*x^5) - (2*B*b*(b*x + c*x^2)^(1/2
))/(9*x^5) - (20*B*c*(b*x + c*x^2)^(1/2))/(63*x^4) - (2*A*c^2*(b*x + c*x^2)^(1/2))/(231*b*x^4) - (2*A*b*(b*x +
c*x^2)^(1/2))/(11*x^6) - (16*A*c^4*(b*x + c*x^2)^(1/2))/(1155*b^3*x^2) + (32*A*c^5*(b*x + c*x^2)^(1/2))/(1155
*b^4*x) - (2*B*c^2*(b*x + c*x^2)^(1/2))/(105*b*x^3) + (8*B*c^3*(b*x + c*x^2)^(1/2))/(315*b^2*x^2) - (16*B*c^4*
(b*x + c*x^2)^(1/2))/(315*b^3*x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**8,x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/x**8, x)

________________________________________________________________________________________