3.82 \(\int x^{7/2} (b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=164 \[ -\frac{512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

[Out]

(-512*b^5*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (256*b^4*(b*x + c*x^2)^(5/2))/(9009*c^5*x^(3/2)) - (64*b^
3*(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (32*b^2*Sqrt[x]*(b*x + c*x^2)^(5/2))/(429*c^3) - (4*b*x^(3/2)*(b*x
+ c*x^2)^(5/2))/(39*c^2) + (2*x^(5/2)*(b*x + c*x^2)^(5/2))/(15*c)

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Rubi [A]  time = 0.075216, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ -\frac{512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(b*x + c*x^2)^(3/2),x]

[Out]

(-512*b^5*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (256*b^4*(b*x + c*x^2)^(5/2))/(9009*c^5*x^(3/2)) - (64*b^
3*(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (32*b^2*Sqrt[x]*(b*x + c*x^2)^(5/2))/(429*c^3) - (4*b*x^(3/2)*(b*x
+ c*x^2)^(5/2))/(39*c^2) + (2*x^(5/2)*(b*x + c*x^2)^(5/2))/(15*c)

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} \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac{(2 b) \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx}{3 c}\\ &=-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}+\frac{\left (16 b^2\right ) \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx}{39 c^2}\\ &=\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac{\left (32 b^3\right ) \int \sqrt{x} \left (b x+c x^2\right )^{3/2} \, dx}{143 c^3}\\ &=-\frac{64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}+\frac{\left (128 b^4\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{1287 c^4}\\ &=\frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac{\left (256 b^5\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{9009 c^5}\\ &=-\frac{512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}\\ \end{align*}

Mathematica [A]  time = 0.0462373, size = 75, normalized size = 0.46 \[ \frac{2 (x (b+c x))^{5/2} \left (-1120 b^3 c^2 x^2+1680 b^2 c^3 x^3+640 b^4 c x-256 b^5-2310 b c^4 x^4+3003 c^5 x^5\right )}{45045 c^6 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(b*x + c*x^2)^(3/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*(-256*b^5 + 640*b^4*c*x - 1120*b^3*c^2*x^2 + 1680*b^2*c^3*x^3 - 2310*b*c^4*x^4 + 3003*c
^5*x^5))/(45045*c^6*x^(5/2))

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Maple [A]  time = 0.054, size = 77, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3003\,{x}^{5}{c}^{5}+2310\,b{x}^{4}{c}^{4}-1680\,{b}^{2}{x}^{3}{c}^{3}+1120\,{b}^{3}{x}^{2}{c}^{2}-640\,{b}^{4}xc+256\,{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^(7/2)*(c*x^2+b*x)^(3/2),x)

[Out]

-2/45045*(c*x+b)*(-3003*c^5*x^5+2310*b*c^4*x^4-1680*b^2*c^3*x^3+1120*b^3*c^2*x^2-640*b^4*c*x+256*b^5)*(c*x^2+b
*x)^(3/2)/c^6/x^(3/2)

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Maxima [A]  time = 1.1401, size = 227, normalized size = 1.38 \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} + 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}}{45045 \, c^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^2+b*x)^(3/2),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 + 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)/(c^6*x^6)

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Fricas [A]  time = 2.04401, size = 223, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (3003 \, c^{7} x^{7} + 3696 \, b c^{6} x^{6} + 63 \, b^{2} c^{5} x^{5} - 70 \, b^{3} c^{4} x^{4} + 80 \, b^{4} c^{3} x^{3} - 96 \, b^{5} c^{2} x^{2} + 128 \, b^{6} c x - 256 \, b^{7}\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)*(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*c^7*x^7 + 3696*b*c^6*x^6 + 63*b^2*c^5*x^5 - 70*b^3*c^4*x^4 + 80*b^4*c^3*x^3 - 96*b^5*c^2*x^2 + 1
28*b^6*c*x - 256*b^7)*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**(7/2)*(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.32596, size = 246, normalized size = 1.5 \begin{align*} -\frac{2}{45045} \, 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{\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

-2/45045*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*(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)