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

Optimal. Leaf size=207 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{765765 c^6 x^{7/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{109395 c^5 x^{5/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{12155 c^4 x^{3/2}}+\frac{16 b \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{3315 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c} \]

[Out]

(-256*b^4*(10*b*B - 17*A*c)*(b*x + c*x^2)^(7/2))/(765765*c^6*x^(7/2)) + (128*b^3*(10*b*B - 17*A*c)*(b*x + c*x^
2)^(7/2))/(109395*c^5*x^(5/2)) - (32*b^2*(10*b*B - 17*A*c)*(b*x + c*x^2)^(7/2))/(12155*c^4*x^(3/2)) + (16*b*(1
0*b*B - 17*A*c)*(b*x + c*x^2)^(7/2))/(3315*c^3*Sqrt[x]) - (2*(10*b*B - 17*A*c)*Sqrt[x]*(b*x + c*x^2)^(7/2))/(2
55*c^2) + (2*B*x^(3/2)*(b*x + c*x^2)^(7/2))/(17*c)

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Rubi [A]  time = 0.203379, 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} \[ -\frac{256 b^4 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{765765 c^6 x^{7/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{109395 c^5 x^{5/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{12155 c^4 x^{3/2}}+\frac{16 b \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{3315 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256*b^4*(10*b*B - 17*A*c)*(b*x + c*x^2)^(7/2))/(765765*c^6*x^(7/2)) + (128*b^3*(10*b*B - 17*A*c)*(b*x + c*x^
2)^(7/2))/(109395*c^5*x^(5/2)) - (32*b^2*(10*b*B - 17*A*c)*(b*x + c*x^2)^(7/2))/(12155*c^4*x^(3/2)) + (16*b*(1
0*b*B - 17*A*c)*(b*x + c*x^2)^(7/2))/(3315*c^3*Sqrt[x]) - (2*(10*b*B - 17*A*c)*Sqrt[x]*(b*x + c*x^2)^(7/2))/(2
55*c^2) + (2*B*x^(3/2)*(b*x + c*x^2)^(7/2))/(17*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^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx &=\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}+\frac{\left (2 \left (\frac{3}{2} (-b B+A c)+\frac{7}{2} (-b B+2 A c)\right )\right ) \int x^{3/2} \left (b x+c x^2\right )^{5/2} \, dx}{17 c}\\ &=-\frac{2 (10 b B-17 A c) \sqrt{x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}+\frac{(8 b (10 b B-17 A c)) \int \sqrt{x} \left (b x+c x^2\right )^{5/2} \, dx}{255 c^2}\\ &=\frac{16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt{x}}-\frac{2 (10 b B-17 A c) \sqrt{x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac{\left (16 b^2 (10 b B-17 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{\sqrt{x}} \, dx}{1105 c^3}\\ &=-\frac{32 b^2 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{12155 c^4 x^{3/2}}+\frac{16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt{x}}-\frac{2 (10 b B-17 A c) \sqrt{x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}+\frac{\left (64 b^3 (10 b B-17 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^{3/2}} \, dx}{12155 c^4}\\ &=\frac{128 b^3 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{109395 c^5 x^{5/2}}-\frac{32 b^2 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{12155 c^4 x^{3/2}}+\frac{16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt{x}}-\frac{2 (10 b B-17 A c) \sqrt{x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac{\left (128 b^4 (10 b B-17 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^{5/2}} \, dx}{109395 c^5}\\ &=-\frac{256 b^4 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{765765 c^6 x^{7/2}}+\frac{128 b^3 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{109395 c^5 x^{5/2}}-\frac{32 b^2 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{12155 c^4 x^{3/2}}+\frac{16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt{x}}-\frac{2 (10 b B-17 A c) \sqrt{x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}\\ \end{align*}

Mathematica [A]  time = 0.100492, size = 120, normalized size = 0.58 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (336 b^2 c^3 x^2 (51 A+55 B x)-224 b^3 c^2 x (34 A+45 B x)+128 b^4 c (17 A+35 B x)-462 b c^4 x^3 (68 A+65 B x)+3003 c^5 x^4 (17 A+15 B x)-1280 b^5 B\right )}{765765 c^6 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(-1280*b^5*B + 3003*c^5*x^4*(17*A + 15*B*x) + 128*b^4*c*(17*A + 35*B*x) - 224
*b^3*c^2*x*(34*A + 45*B*x) + 336*b^2*c^3*x^2*(51*A + 55*B*x) - 462*b*c^4*x^3*(68*A + 65*B*x)))/(765765*c^6*Sqr
t[x])

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Maple [A]  time = 0.004, size = 131, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 45045\,B{x}^{5}{c}^{5}+51051\,A{c}^{5}{x}^{4}-30030\,Bb{c}^{4}{x}^{4}-31416\,Ab{c}^{4}{x}^{3}+18480\,B{b}^{2}{c}^{3}{x}^{3}+17136\,A{b}^{2}{c}^{3}{x}^{2}-10080\,B{b}^{3}{c}^{2}{x}^{2}-7616\,A{b}^{3}{c}^{2}x+4480\,B{b}^{4}cx+2176\,A{b}^{4}c-1280\,B{b}^{5} \right ) }{765765\,{c}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(c*x+b)*(45045*B*c^5*x^5+51051*A*c^5*x^4-30030*B*b*c^4*x^4-31416*A*b*c^4*x^3+18480*B*b^2*c^3*x^3+1713
6*A*b^2*c^3*x^2-10080*B*b^3*c^2*x^2-7616*A*b^3*c^2*x+4480*B*b^4*c*x+2176*A*b^4*c-1280*B*b^5)*(c*x^2+b*x)^(5/2)
/c^6/x^(5/2)

________________________________________________________________________________________

Maxima [B]  time = 1.16114, size = 684, normalized size = 3.3 \begin{align*} \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} + 10 \,{\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} + 13 \,{\left (315 \, b^{2} c^{5} x^{7} + 35 \, b^{3} c^{4} x^{6} - 40 \, b^{4} c^{3} x^{5} + 48 \, b^{5} c^{2} x^{4} - 64 \, b^{6} c x^{3} + 128 \, b^{7} x^{2}\right )} x^{4}\right )} \sqrt{c x + b} A}{45045 \, c^{5} x^{6}} + \frac{2 \,{\left (7 \,{\left (6435 \, c^{8} x^{8} + 429 \, b c^{7} x^{7} - 462 \, b^{2} c^{6} x^{6} + 504 \, b^{3} c^{5} x^{5} - 560 \, b^{4} c^{4} x^{4} + 640 \, b^{5} c^{3} x^{3} - 768 \, b^{6} c^{2} x^{2} + 1024 \, b^{7} c x - 2048 \, b^{8}\right )} x^{7} + 34 \,{\left (3003 \, b c^{7} x^{8} + 231 \, b^{2} c^{6} x^{7} - 252 \, b^{3} c^{5} x^{6} + 280 \, b^{4} c^{4} x^{5} - 320 \, b^{5} c^{3} x^{4} + 384 \, b^{6} c^{2} x^{3} - 512 \, b^{7} c x^{2} + 1024 \, b^{8} x\right )} x^{6} + 85 \,{\left (693 \, b^{2} c^{6} x^{8} + 63 \, b^{3} c^{5} x^{7} - 70 \, b^{4} c^{4} x^{6} + 80 \, b^{5} c^{3} x^{5} - 96 \, b^{6} c^{2} x^{4} + 128 \, b^{7} c x^{3} - 256 \, b^{8} x^{2}\right )} x^{5}\right )} \sqrt{c x + b} B}{765765 \, c^{6} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 - 320*b^4*c^3*x^3 + 384*b^5*c^2*x^2
- 512*b^6*c*x + 1024*b^7)*x^6 + 10*(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 + 13*(315*b^2*c^5*x^7 + 35*b^3*c^4*x^6 - 40*b^4*c^3*x^5 + 48*b^5*c^2
*x^4 - 64*b^6*c*x^3 + 128*b^7*x^2)*x^4)*sqrt(c*x + b)*A/(c^5*x^6) + 2/765765*(7*(6435*c^8*x^8 + 429*b*c^7*x^7
- 462*b^2*c^6*x^6 + 504*b^3*c^5*x^5 - 560*b^4*c^4*x^4 + 640*b^5*c^3*x^3 - 768*b^6*c^2*x^2 + 1024*b^7*c*x - 204
8*b^8)*x^7 + 34*(3003*b*c^7*x^8 + 231*b^2*c^6*x^7 - 252*b^3*c^5*x^6 + 280*b^4*c^4*x^5 - 320*b^5*c^3*x^4 + 384*
b^6*c^2*x^3 - 512*b^7*c*x^2 + 1024*b^8*x)*x^6 + 85*(693*b^2*c^6*x^8 + 63*b^3*c^5*x^7 - 70*b^4*c^4*x^6 + 80*b^5
*c^3*x^5 - 96*b^6*c^2*x^4 + 128*b^7*c*x^3 - 256*b^8*x^2)*x^5)*sqrt(c*x + b)*B/(c^6*x^7)

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Fricas [A]  time = 1.596, size = 479, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (45045 \, B c^{8} x^{8} - 1280 \, B b^{8} + 2176 \, A b^{7} c + 3003 \,{\left (35 \, B b c^{7} + 17 \, A c^{8}\right )} x^{7} + 231 \,{\left (275 \, B b^{2} c^{6} + 527 \, A b c^{7}\right )} x^{6} + 63 \,{\left (5 \, B b^{3} c^{5} + 1207 \, A b^{2} c^{6}\right )} x^{5} - 35 \,{\left (10 \, B b^{4} c^{4} - 17 \, A b^{3} c^{5}\right )} x^{4} + 40 \,{\left (10 \, B b^{5} c^{3} - 17 \, A b^{4} c^{4}\right )} x^{3} - 48 \,{\left (10 \, B b^{6} c^{2} - 17 \, A b^{5} c^{3}\right )} x^{2} + 64 \,{\left (10 \, B b^{7} c - 17 \, A b^{6} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{765765 \, c^{6} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(45045*B*c^8*x^8 - 1280*B*b^8 + 2176*A*b^7*c + 3003*(35*B*b*c^7 + 17*A*c^8)*x^7 + 231*(275*B*b^2*c^6
+ 527*A*b*c^7)*x^6 + 63*(5*B*b^3*c^5 + 1207*A*b^2*c^6)*x^5 - 35*(10*B*b^4*c^4 - 17*A*b^3*c^5)*x^4 + 40*(10*B*b
^5*c^3 - 17*A*b^4*c^4)*x^3 - 48*(10*B*b^6*c^2 - 17*A*b^5*c^3)*x^2 + 64*(10*B*b^7*c - 17*A*b^6*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**(3/2)*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.20946, size = 756, normalized size = 3.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/109395*B*c^2*(2048*b^(17/2)/c^8 + (6435*(c*x + b)^(17/2) - 51051*(c*x + b)^(15/2)*b + 176715*(c*x + b)^(13/2
)*b^2 - 348075*(c*x + b)^(11/2)*b^3 + 425425*(c*x + b)^(9/2)*b^4 - 328185*(c*x + b)^(7/2)*b^5 + 153153*(c*x +
b)^(5/2)*b^6 - 36465*(c*x + b)^(3/2)*b^7)/c^8) - 4/45045*B*b*c*(1024*b^(15/2)/c^7 - (3003*(c*x + b)^(15/2) - 2
0790*(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) - 2/45045*A*c^2*(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^2*(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 + 90
09*(c*x + b)^(5/2)*b^4 - 3003*(c*x + b)^(3/2)*b^5)/c^6) + 4/9009*A*b*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^2*(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)