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

Optimal. Leaf size=207 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt{x}}-\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \]

[Out]

(-256*b^4*(10*b*B - 13*A*c)*(b*x + c*x^2)^(3/2))/(45045*c^6*x^(3/2)) + (128*b^3*(10*b*B - 13*A*c)*(b*x + c*x^2
)^(3/2))/(15015*c^5*Sqrt[x]) - (32*b^2*(10*b*B - 13*A*c)*Sqrt[x]*(b*x + c*x^2)^(3/2))/(3003*c^4) + (16*b*(10*b
*B - 13*A*c)*x^(3/2)*(b*x + c*x^2)^(3/2))/(1287*c^3) - (2*(10*b*B - 13*A*c)*x^(5/2)*(b*x + c*x^2)^(3/2))/(143*
c^2) + (2*B*x^(7/2)*(b*x + c*x^2)^(3/2))/(13*c)

________________________________________________________________________________________

Rubi [A]  time = 0.194197, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {794, 656, 648} \[ -\frac{256 b^4 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt{x}}-\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256*b^4*(10*b*B - 13*A*c)*(b*x + c*x^2)^(3/2))/(45045*c^6*x^(3/2)) + (128*b^3*(10*b*B - 13*A*c)*(b*x + c*x^2
)^(3/2))/(15015*c^5*Sqrt[x]) - (32*b^2*(10*b*B - 13*A*c)*Sqrt[x]*(b*x + c*x^2)^(3/2))/(3003*c^4) + (16*b*(10*b
*B - 13*A*c)*x^(3/2)*(b*x + c*x^2)^(3/2))/(1287*c^3) - (2*(10*b*B - 13*A*c)*x^(5/2)*(b*x + c*x^2)^(3/2))/(143*
c^2) + (2*B*x^(7/2)*(b*x + c*x^2)^(3/2))/(13*c)

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), 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 + p, 0]

Rubi steps

\begin{align*} \int x^{7/2} (A+B x) \sqrt{b x+c x^2} \, dx &=\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac{\left (2 \left (\frac{7}{2} (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int x^{7/2} \sqrt{b x+c x^2} \, dx}{13 c}\\ &=-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac{(8 b (10 b B-13 A c)) \int x^{5/2} \sqrt{b x+c x^2} \, dx}{143 c^2}\\ &=\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac{\left (16 b^2 (10 b B-13 A c)\right ) \int x^{3/2} \sqrt{b x+c x^2} \, dx}{429 c^3}\\ &=-\frac{32 b^2 (10 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac{\left (64 b^3 (10 b B-13 A c)\right ) \int \sqrt{x} \sqrt{b x+c x^2} \, dx}{3003 c^4}\\ &=\frac{128 b^3 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{15015 c^5 \sqrt{x}}-\frac{32 b^2 (10 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac{\left (128 b^4 (10 b B-13 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{15015 c^5}\\ &=-\frac{256 b^4 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{45045 c^6 x^{3/2}}+\frac{128 b^3 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{15015 c^5 \sqrt{x}}-\frac{32 b^2 (10 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}\\ \end{align*}

Mathematica [A]  time = 0.0952725, size = 113, normalized size = 0.55 \[ \frac{2 (x (b+c x))^{3/2} \left (80 b^2 c^3 x^2 (39 A+35 B x)-96 b^3 c^2 x (26 A+25 B x)+128 b^4 c (13 A+15 B x)-70 b c^4 x^3 (52 A+45 B x)+315 c^5 x^4 (13 A+11 B x)-1280 b^5 B\right )}{45045 c^6 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x*(b + c*x))^(3/2)*(-1280*b^5*B + 315*c^5*x^4*(13*A + 11*B*x) + 128*b^4*c*(13*A + 15*B*x) - 96*b^3*c^2*x*(
26*A + 25*B*x) + 80*b^2*c^3*x^2*(39*A + 35*B*x) - 70*b*c^4*x^3*(52*A + 45*B*x)))/(45045*c^6*x^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 131, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3465\,B{x}^{5}{c}^{5}+4095\,A{c}^{5}{x}^{4}-3150\,Bb{c}^{4}{x}^{4}-3640\,Ab{c}^{4}{x}^{3}+2800\,B{b}^{2}{c}^{3}{x}^{3}+3120\,A{b}^{2}{c}^{3}{x}^{2}-2400\,B{b}^{3}{c}^{2}{x}^{2}-2496\,A{b}^{3}{c}^{2}x+1920\,B{b}^{4}cx+1664\,A{b}^{4}c-1280\,B{b}^{5} \right ) }{45045\,{c}^{6}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(c*x+b)*(3465*B*c^5*x^5+4095*A*c^5*x^4-3150*B*b*c^4*x^4-3640*A*b*c^4*x^3+2800*B*b^2*c^3*x^3+3120*A*b^2
*c^3*x^2-2400*B*b^3*c^2*x^2-2496*A*b^3*c^2*x+1920*B*b^4*c*x+1664*A*b^4*c-1280*B*b^5)*(c*x^2+b*x)^(1/2)/c^6/x^(
1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.30937, size = 192, normalized size = 0.93 \begin{align*} \frac{2 \,{\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x + b} A}{3465 \, c^{5}} + \frac{2 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} \sqrt{c x + b} B}{9009 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*c^5*x^5 + 35*b*c^4*x^4 - 40*b^2*c^3*x^3 + 48*b^3*c^2*x^2 - 64*b^4*c*x + 128*b^5)*sqrt(c*x + b)*A/c
^5 + 2/9009*(693*c^6*x^6 + 63*b*c^5*x^5 - 70*b^2*c^4*x^4 + 80*b^3*c^3*x^3 - 96*b^4*c^2*x^2 + 128*b^5*c*x - 256
*b^6)*sqrt(c*x + b)*B/c^6

________________________________________________________________________________________

Fricas [A]  time = 1.62563, size = 360, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (3465 \, B c^{6} x^{6} - 1280 \, B b^{6} + 1664 \, A b^{5} c + 315 \,{\left (B b c^{5} + 13 \, A c^{6}\right )} x^{5} - 35 \,{\left (10 \, B b^{2} c^{4} - 13 \, A b c^{5}\right )} x^{4} + 40 \,{\left (10 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} x^{3} - 48 \,{\left (10 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{2} + 64 \,{\left (10 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{45045 \, c^{6} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*B*c^6*x^6 - 1280*B*b^6 + 1664*A*b^5*c + 315*(B*b*c^5 + 13*A*c^6)*x^5 - 35*(10*B*b^2*c^4 - 13*A*b
*c^5)*x^4 + 40*(10*B*b^3*c^3 - 13*A*b^2*c^4)*x^3 - 48*(10*B*b^4*c^2 - 13*A*b^3*c^3)*x^2 + 64*(10*B*b^5*c - 13*
A*b^4*c^2)*x)*sqrt(c*x^2 + b*x)/(c^6*sqrt(x))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.1508, size = 213, normalized size = 1.03 \begin{align*} \frac{2}{9009} \, B{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} - \frac{2}{3465} \, A{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*B*(256*b^(13/2)/c^6 + (693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 10010*(c*x + b)^(9/2)*b^2 - 128
70*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4 - 3003*(c*x + b)^(3/2)*b^5)/c^6) - 2/3465*A*(128*b^(11/2)/c^
5 - (315*(c*x + b)^(11/2) - 1540*(c*x + b)^(9/2)*b + 2970*(c*x + b)^(7/2)*b^2 - 2772*(c*x + b)^(5/2)*b^3 + 115
5*(c*x + b)^(3/2)*b^4)/c^5)