3.300 \(\int x^3 (a+b x)^{5/2} \, dx\)

Optimal. Leaf size=72 \[ \frac{2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac{2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac{2 (a+b x)^{13/2}}{13 b^4}-\frac{6 a (a+b x)^{11/2}}{11 b^4} \]

[Out]

(-2*a^3*(a + b*x)^(7/2))/(7*b^4) + (2*a^2*(a + b*x)^(9/2))/(3*b^4) - (6*a*(a + b*x)^(11/2))/(11*b^4) + (2*(a +
b*x)^(13/2))/(13*b^4)

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Rubi [A]  time = 0.0171271, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac{2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac{2 (a+b x)^{13/2}}{13 b^4}-\frac{6 a (a+b x)^{11/2}}{11 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(5/2),x]

[Out]

(-2*a^3*(a + b*x)^(7/2))/(7*b^4) + (2*a^2*(a + b*x)^(9/2))/(3*b^4) - (6*a*(a + b*x)^(11/2))/(11*b^4) + (2*(a +
b*x)^(13/2))/(13*b^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^3 (a+b x)^{5/2} \, dx &=\int \left (-\frac{a^3 (a+b x)^{5/2}}{b^3}+\frac{3 a^2 (a+b x)^{7/2}}{b^3}-\frac{3 a (a+b x)^{9/2}}{b^3}+\frac{(a+b x)^{11/2}}{b^3}\right ) \, dx\\ &=-\frac{2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac{2 a^2 (a+b x)^{9/2}}{3 b^4}-\frac{6 a (a+b x)^{11/2}}{11 b^4}+\frac{2 (a+b x)^{13/2}}{13 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0385916, size = 46, normalized size = 0.64 \[ \frac{2 (a+b x)^{7/2} \left (56 a^2 b x-16 a^3-126 a b^2 x^2+231 b^3 x^3\right )}{3003 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2)*(-16*a^3 + 56*a^2*b*x - 126*a*b^2*x^2 + 231*b^3*x^3))/(3003*b^4)

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Maple [A]  time = 0.005, size = 43, normalized size = 0.6 \begin{align*} -{\frac{-462\,{b}^{3}{x}^{3}+252\,a{b}^{2}{x}^{2}-112\,{a}^{2}bx+32\,{a}^{3}}{3003\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(5/2),x)

[Out]

-2/3003*(b*x+a)^(7/2)*(-231*b^3*x^3+126*a*b^2*x^2-56*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.08305, size = 76, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{13}{2}}}{13 \, b^{4}} - \frac{6 \,{\left (b x + a\right )}^{\frac{11}{2}} a}{11 \, b^{4}} + \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2}}{3 \, b^{4}} - \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3}}{7 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*(b*x + a)^(13/2)/b^4 - 6/11*(b*x + a)^(11/2)*a/b^4 + 2/3*(b*x + a)^(9/2)*a^2/b^4 - 2/7*(b*x + a)^(7/2)*a^
3/b^4

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Fricas [A]  time = 1.54731, size = 171, normalized size = 2.38 \begin{align*} \frac{2 \,{\left (231 \, b^{6} x^{6} + 567 \, a b^{5} x^{5} + 371 \, a^{2} b^{4} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{4} b^{2} x^{2} + 8 \, a^{5} b x - 16 \, a^{6}\right )} \sqrt{b x + a}}{3003 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*b^6*x^6 + 567*a*b^5*x^5 + 371*a^2*b^4*x^4 + 5*a^3*b^3*x^3 - 6*a^4*b^2*x^2 + 8*a^5*b*x - 16*a^6)*sq
rt(b*x + a)/b^4

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Sympy [A]  time = 5.89415, size = 146, normalized size = 2.03 \begin{align*} \begin{cases} - \frac{32 a^{6} \sqrt{a + b x}}{3003 b^{4}} + \frac{16 a^{5} x \sqrt{a + b x}}{3003 b^{3}} - \frac{4 a^{4} x^{2} \sqrt{a + b x}}{1001 b^{2}} + \frac{10 a^{3} x^{3} \sqrt{a + b x}}{3003 b} + \frac{106 a^{2} x^{4} \sqrt{a + b x}}{429} + \frac{54 a b x^{5} \sqrt{a + b x}}{143} + \frac{2 b^{2} x^{6} \sqrt{a + b x}}{13} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**(5/2),x)

[Out]

Piecewise((-32*a**6*sqrt(a + b*x)/(3003*b**4) + 16*a**5*x*sqrt(a + b*x)/(3003*b**3) - 4*a**4*x**2*sqrt(a + b*x
)/(1001*b**2) + 10*a**3*x**3*sqrt(a + b*x)/(3003*b) + 106*a**2*x**4*sqrt(a + b*x)/429 + 54*a*b*x**5*sqrt(a + b
*x)/143 + 2*b**2*x**6*sqrt(a + b*x)/13, Ne(b, 0)), (a**(5/2)*x**4/4, True))

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Giac [B]  time = 1.19315, size = 261, normalized size = 3.62 \begin{align*} \frac{2 \,{\left (\frac{143 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} a^{2}}{b^{3}} + \frac{26 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )} a}{b^{3}} + \frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5}\right )}}{b^{3}}\right )}}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(143*(35*(b*x + a)^(9/2) - 135*(b*x + a)^(7/2)*a + 189*(b*x + a)^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3)*
a^2/b^3 + 26*(315*(b*x + a)^(11/2) - 1540*(b*x + a)^(9/2)*a + 2970*(b*x + a)^(7/2)*a^2 - 2772*(b*x + a)^(5/2)*
a^3 + 1155*(b*x + a)^(3/2)*a^4)*a/b^3 + 5*(693*(b*x + a)^(13/2) - 4095*(b*x + a)^(11/2)*a + 10010*(b*x + a)^(9
/2)*a^2 - 12870*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 3003*(b*x + a)^(3/2)*a^5)/b^3)/b