3.203 \(\int x^{5/2} (A+B x) (b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=207 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{45045 c^6 x^{5/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{9009 c^5 x^{3/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{1287 c^4 \sqrt{x}}+\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{429 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

[Out]

(-256*b^4*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (128*b^3*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5
/2))/(9009*c^5*x^(3/2)) - (32*b^2*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (16*b*(2*b*B - 3*A
*c)*Sqrt[x]*(b*x + c*x^2)^(5/2))/(429*c^3) - (2*(2*b*B - 3*A*c)*x^(3/2)*(b*x + c*x^2)^(5/2))/(39*c^2) + (2*B*x
^(5/2)*(b*x + c*x^2)^(5/2))/(15*c)

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Rubi [A]  time = 0.153077, 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} \[ -\frac{256 b^4 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{45045 c^6 x^{5/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{9009 c^5 x^{3/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{1287 c^4 \sqrt{x}}+\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{429 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(b*x + c*x^2)^(3/2),x]

[Out]

(-256*b^4*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (128*b^3*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5
/2))/(9009*c^5*x^(3/2)) - (32*b^2*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (16*b*(2*b*B - 3*A
*c)*Sqrt[x]*(b*x + c*x^2)^(5/2))/(429*c^3) - (2*(2*b*B - 3*A*c)*x^(3/2)*(b*x + c*x^2)^(5/2))/(39*c^2) + (2*B*x
^(5/2)*(b*x + c*x^2)^(5/2))/(15*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^{5/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}+\frac{\left (2 \left (\frac{5}{2} (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right )\right ) \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx}{15 c}\\ &=-\frac{2 (2 b B-3 A c) x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}+\frac{(8 b (2 b B-3 A c)) \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx}{39 c^2}\\ &=\frac{16 b (2 b B-3 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{2 (2 b B-3 A c) x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac{\left (16 b^2 (2 b B-3 A c)\right ) \int \sqrt{x} \left (b x+c x^2\right )^{3/2} \, dx}{143 c^3}\\ &=-\frac{32 b^2 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{16 b (2 b B-3 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{2 (2 b B-3 A c) x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}+\frac{\left (64 b^3 (2 b B-3 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{1287 c^4}\\ &=\frac{128 b^3 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{32 b^2 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{16 b (2 b B-3 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{2 (2 b B-3 A c) x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac{\left (128 b^4 (2 b B-3 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{9009 c^5}\\ &=-\frac{256 b^4 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac{128 b^3 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{32 b^2 (2 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{16 b (2 b B-3 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{2 (2 b B-3 A c) x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}\\ \end{align*}

Mathematica [A]  time = 0.0971498, size = 110, normalized size = 0.53 \[ \frac{2 (x (b+c x))^{5/2} \left (1680 b^2 c^3 x^2 (A+B x)-160 b^3 c^2 x (6 A+7 B x)+128 b^4 c (3 A+5 B x)-210 b c^4 x^3 (12 A+11 B x)+231 c^5 x^4 (15 A+13 B x)-256 b^5 B\right )}{45045 c^6 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x)*(b*x + c*x^2)^(3/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*(-256*b^5*B + 1680*b^2*c^3*x^2*(A + B*x) + 128*b^4*c*(3*A + 5*B*x) - 160*b^3*c^2*x*(6*A
+ 7*B*x) - 210*b*c^4*x^3*(12*A + 11*B*x) + 231*c^5*x^4*(15*A + 13*B*x)))/(45045*c^6*x^(5/2))

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Maple [A]  time = 0.007, size = 131, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3003\,B{x}^{5}{c}^{5}+3465\,A{c}^{5}{x}^{4}-2310\,Bb{c}^{4}{x}^{4}-2520\,Ab{c}^{4}{x}^{3}+1680\,B{b}^{2}{c}^{3}{x}^{3}+1680\,A{b}^{2}{c}^{3}{x}^{2}-1120\,B{b}^{3}{c}^{2}{x}^{2}-960\,A{b}^{3}{c}^{2}x+640\,B{b}^{4}cx+384\,A{b}^{4}c-256\,B{b}^{5} \right ) }{45045\,{c}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)*(c*x^2+b*x)^(3/2),x)

[Out]

2/45045*(c*x+b)*(3003*B*c^5*x^5+3465*A*c^5*x^4-2310*B*b*c^4*x^4-2520*A*b*c^4*x^3+1680*B*b^2*c^3*x^3+1680*A*b^2
*c^3*x^2-1120*B*b^3*c^2*x^2-960*A*b^3*c^2*x+640*B*b^4*c*x+384*A*b^4*c-256*B*b^5)*(c*x^2+b*x)^(3/2)/c^6/x^(3/2)

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Maxima [A]  time = 1.18542, size = 429, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (5 \,{\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 )} x^{5} + 13 \,{\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt{c x + b} A}{45045 \, c^{5} x^{5}} + \frac{2 \,{\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 5 \,{\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5}\right )} \sqrt{c x + b} B}{45045 \, c^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

2/45045*(5*(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)*x^5 + 13*(315*b*c^5*x^6 + 35*b^2*c^4*x^5 - 40*b^3*c^3*x^4 + 48*b^4*c^2*x^3 - 64*b^5*c*x^2 + 128*b^6*x)*x^
4)*sqrt(c*x + b)*A/(c^5*x^5) + 2/45045*((3003*c^7*x^7 + 231*b*c^6*x^6 - 252*b^2*c^5*x^5 + 280*b^3*c^4*x^4 - 32
0*b^4*c^3*x^3 + 384*b^5*c^2*x^2 - 512*b^6*c*x + 1024*b^7)*x^6 + 5*(693*b*c^6*x^7 + 63*b^2*c^5*x^6 - 70*b^3*c^4
*x^5 + 80*b^4*c^3*x^4 - 96*b^5*c^2*x^3 + 128*b^6*c*x^2 - 256*b^7*x)*x^5)*sqrt(c*x + b)*B/(c^6*x^6)

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Fricas [A]  time = 1.4611, size = 400, normalized size = 1.93 \begin{align*} \frac{2 \,{\left (3003 \, B c^{7} x^{7} - 256 \, B b^{7} + 384 \, A b^{6} c + 231 \,{\left (16 \, B b c^{6} + 15 \, A c^{7}\right )} x^{6} + 63 \,{\left (B b^{2} c^{5} + 70 \, A b c^{6}\right )} x^{5} - 35 \,{\left (2 \, B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} x^{4} + 40 \,{\left (2 \, B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} x^{3} - 48 \,{\left (2 \, B b^{5} c^{2} - 3 \, A b^{4} c^{3}\right )} x^{2} + 64 \,{\left (2 \, B b^{6} c - 3 \, A b^{5} 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^(5/2)*(B*x+A)*(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^7*x^7 - 256*B*b^7 + 384*A*b^6*c + 231*(16*B*b*c^6 + 15*A*c^7)*x^6 + 63*(B*b^2*c^5 + 70*A*b*c
^6)*x^5 - 35*(2*B*b^3*c^4 - 3*A*b^2*c^5)*x^4 + 40*(2*B*b^4*c^3 - 3*A*b^3*c^4)*x^3 - 48*(2*B*b^5*c^2 - 3*A*b^4*
c^3)*x^2 + 64*(2*B*b^6*c - 3*A*b^5*c^2)*x)*sqrt(c*x^2 + b*x)/(c^6*sqrt(x))

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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**(5/2)*(B*x+A)*(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.20355, size = 463, normalized size = 2.24 \begin{align*} -\frac{2}{45045} \, B c{\left (\frac{1024 \, b^{\frac{15}{2}}}{c^{7}} - \frac{3003 \,{\left (c x + b\right )}^{\frac{15}{2}} - 20790 \,{\left (c x + b\right )}^{\frac{13}{2}} b + 61425 \,{\left (c x + b\right )}^{\frac{11}{2}} b^{2} - 100100 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{3} + 96525 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{4} - 54054 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{5} + 15015 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{6}}{c^{7}}\right )} + \frac{2}{9009} \, B 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}{9009} \, A c{\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 b{\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^(5/2)*(B*x+A)*(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

-2/45045*B*c*(1024*b^(15/2)/c^7 - (3003*(c*x + b)^(15/2) - 20790*(c*x + b)^(13/2)*b + 61425*(c*x + b)^(11/2)*b
^2 - 100100*(c*x + b)^(9/2)*b^3 + 96525*(c*x + b)^(7/2)*b^4 - 54054*(c*x + b)^(5/2)*b^5 + 15015*(c*x + b)^(3/2
)*b^6)/c^7) + 2/9009*B*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 - 12870*(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) + 2/9009*A*
c*(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 - 12870*(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*b*(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 + 1155*(c*
x + b)^(3/2)*b^4)/c^5)