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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 = 0.03, size = 46, normalized size = 0.64 \[ \frac {2 (a+b x)^{7/2} \left (-16 a^3+56 a^2 b x-126 a b^2 x^2+231 b^3 x^3\right )}{3003 b^4} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 75, normalized size = 1.04 \[ \frac {2 \, {\left (231 \, b^{6} x^{6} + 567 \, a b^{5} x^{5} + 371 \, a^{2} b^{4} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{4} b^{2} x^{2} + 8 \, a^{5} b x - 16 \, a^{6}\right )} \sqrt {b x + a}}{3003 \, b^{4}} \]

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

________________________________________________________________________________________

giac [B]  time = 1.16, size = 281, normalized size = 3.90 \[ \frac {2 \, {\left (\frac {429 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{3}}{b^{3}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a^{2}}{b^{3}} + \frac {65 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a}{b^{3}} + \frac {5 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )}}{b^{3}}\right )}}{15015 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(429*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*a^3/b^
3 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*
sqrt(b*x + a)*a^4)*a^2/b^3 + 65*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*
(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*a/b^3 + 5*(231*(b*x + a)^(13/2) - 1638
*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b
*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)/b^3)/b

________________________________________________________________________________________

maple [A]  time = 0.01, size = 43, normalized size = 0.60 \[ -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-231 b^{3} x^{3}+126 a \,b^{2} x^{2}-56 a^{2} b x +16 a^{3}\right )}{3003 b^{4}} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.35, size = 56, normalized size = 0.78 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {13}{2}}}{13 \, b^{4}} - \frac {6 \, {\left (b x + a\right )}^{\frac {11}{2}} a}{11 \, b^{4}} + \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2}}{3 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{4}} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 56, normalized size = 0.78 \[ \frac {2\,{\left (a+b\,x\right )}^{13/2}}{13\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{9/2}}{3\,b^4}-\frac {6\,a\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^(5/2),x)

[Out]

(2*(a + b*x)^(13/2))/(13*b^4) - (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)

________________________________________________________________________________________

sympy [A]  time = 4.47, size = 146, normalized size = 2.03 \[ \begin {cases} - \frac {32 a^{6} \sqrt {a + b x}}{3003 b^{4}} + \frac {16 a^{5} x \sqrt {a + b x}}{3003 b^{3}} - \frac {4 a^{4} x^{2} \sqrt {a + b x}}{1001 b^{2}} + \frac {10 a^{3} x^{3} \sqrt {a + b x}}{3003 b} + \frac {106 a^{2} x^{4} \sqrt {a + b x}}{429} + \frac {54 a b x^{5} \sqrt {a + b x}}{143} + \frac {2 b^{2} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________