3.3.92 \(\int x^3 (a+b x)^{3/2} \, dx\) [292]

Optimal. Leaf size=72 \[ -\frac {2 a^3 (a+b x)^{5/2}}{5 b^4}+\frac {6 a^2 (a+b x)^{7/2}}{7 b^4}-\frac {2 a (a+b x)^{9/2}}{3 b^4}+\frac {2 (a+b x)^{11/2}}{11 b^4} \]

[Out]

-2/5*a^3*(b*x+a)^(5/2)/b^4+6/7*a^2*(b*x+a)^(7/2)/b^4-2/3*a*(b*x+a)^(9/2)/b^4+2/11*(b*x+a)^(11/2)/b^4

________________________________________________________________________________________

Rubi [A]
time = 0.01, 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 = {45} \begin {gather*} -\frac {2 a^3 (a+b x)^{5/2}}{5 b^4}+\frac {6 a^2 (a+b x)^{7/2}}{7 b^4}+\frac {2 (a+b x)^{11/2}}{11 b^4}-\frac {2 a (a+b x)^{9/2}}{3 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(3/2),x]

[Out]

(-2*a^3*(a + b*x)^(5/2))/(5*b^4) + (6*a^2*(a + b*x)^(7/2))/(7*b^4) - (2*a*(a + b*x)^(9/2))/(3*b^4) + (2*(a + b
*x)^(11/2))/(11*b^4)

Rule 45

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 46, normalized size = 0.64 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (-16 a^3+40 a^2 b x-70 a b^2 x^2+105 b^3 x^3\right )}{1155 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2)*(-16*a^3 + 40*a^2*b*x - 70*a*b^2*x^2 + 105*b^3*x^3))/(1155*b^4)

________________________________________________________________________________________

Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(338\) vs. \(2(72)=144\).
time = 16.41, size = 316, normalized size = 4.39 \begin {gather*} \frac {2 \sqrt {a} \left (16 a^{11} \left (1-\sqrt {\frac {a+b x}{a}}\right )+8 a^{10} b x \left (12-11 \sqrt {\frac {a+b x}{a}}\right )+6 a^9 b^2 x^2 \left (40-33 \sqrt {\frac {a+b x}{a}}\right )+a^8 b^3 x^3 \left (320-231 \sqrt {\frac {a+b x}{a}}\right )+240 a^7 b^4 x^4+924 a^5 b^5 x^5 \left (a+3 b x\right ) \sqrt {\frac {a+b x}{a}}+96 a^6 b^5 x^5+16 a^5 b^6 x^6+66 a^3 b^7 x^7 \left (67 a+64 b x\right ) \sqrt {\frac {a+b x}{a}}+2420 a^2 b^9 x^9 \sqrt {\frac {a+b x}{a}}+770 a b^{10} x^{10} \sqrt {\frac {a+b x}{a}}+105 b^{11} x^{11} \sqrt {\frac {a+b x}{a}}\right )}{1155 b^4 \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^3*(a + b*x)^(3/2),x]')

[Out]

2 Sqrt[a] (16 a ^ 11 (1 - Sqrt[(a + b x) / a]) + 8 a ^ 10 b x (12 - 11 Sqrt[(a + b x) / a]) + 6 a ^ 9 b ^ 2 x
^ 2 (40 - 33 Sqrt[(a + b x) / a]) + a ^ 8 b ^ 3 x ^ 3 (320 - 231 Sqrt[(a + b x) / a]) + 240 a ^ 7 b ^ 4 x ^ 4
+ 924 a ^ 5 b ^ 5 x ^ 5 (a + 3 b x) Sqrt[(a + b x) / a] + 96 a ^ 6 b ^ 5 x ^ 5 + 16 a ^ 5 b ^ 6 x ^ 6 + 66 a ^
3 b ^ 7 x ^ 7 (67 a + 64 b x) Sqrt[(a + b x) / a] + 2420 a ^ 2 b ^ 9 x ^ 9 Sqrt[(a + b x) / a] + 770 a b ^ 10
x ^ 10 Sqrt[(a + b x) / a] + 105 b ^ 11 x ^ 11 Sqrt[(a + b x) / a]) / (1155 b ^ 4 (a ^ 6 + 6 a ^ 5 b x + 15 a
^ 4 b ^ 2 x ^ 2 + 20 a ^ 3 b ^ 3 x ^ 3 + 15 a ^ 2 b ^ 4 x ^ 4 + 6 a b ^ 5 x ^ 5 + b ^ 6 x ^ 6))

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 50, normalized size = 0.69

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-105 b^{3} x^{3}+70 a \,b^{2} x^{2}-40 a^{2} b x +16 a^{3}\right )}{1155 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {2 a \left (b x +a \right )^{\frac {9}{2}}}{3}+\frac {6 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a^{3} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {2 a \left (b x +a \right )^{\frac {9}{2}}}{3}+\frac {6 a^{2} \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {2 a^{3} \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{4}}\)	\(50\)
trager	\(-\frac {2 \left (-105 b^{5} x^{5}-140 a \,b^{4} x^{4}-5 a^{2} b^{3} x^{3}+6 a^{3} b^{2} x^{2}-8 a^{4} b x +16 a^{5}\right ) \sqrt {b x +a}}{1155 b^{4}}\)	\(65\)
risch	\(-\frac {2 \left (-105 b^{5} x^{5}-140 a \,b^{4} x^{4}-5 a^{2} b^{3} x^{3}+6 a^{3} b^{2} x^{2}-8 a^{4} b x +16 a^{5}\right ) \sqrt {b x +a}}{1155 b^{4}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/b^4*(1/11*(b*x+a)^(11/2)-1/3*a*(b*x+a)^(9/2)+3/7*a^2*(b*x+a)^(7/2)-1/5*a^3*(b*x+a)^(5/2))

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 56, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}}}{11 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}} a}{3 \, b^{4}} + \frac {6 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3}}{5 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b*x + a)^(11/2)/b^4 - 2/3*(b*x + a)^(9/2)*a/b^4 + 6/7*(b*x + a)^(7/2)*a^2/b^4 - 2/5*(b*x + a)^(5/2)*a^3/
b^4

________________________________________________________________________________________

Fricas [A]
time = 0.71, size = 64, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (105 \, b^{5} x^{5} + 140 \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{3} - 6 \, a^{3} b^{2} x^{2} + 8 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x + a}}{1155 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*b^5*x^5 + 140*a*b^4*x^4 + 5*a^2*b^3*x^3 - 6*a^3*b^2*x^2 + 8*a^4*b*x - 16*a^5)*sqrt(b*x + a)/b^4

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (68) = 136\).
time = 1.37, size = 1742, normalized size = 24.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32*a**(51/2)*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*
x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 32*a**(51/2)/(1155*a**20*b**4 +
6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x
**5 + 1155*a**14*b**10*x**6) - 176*a**(49/2)*b*x*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*
a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6
) + 192*a**(49/2)*b*x/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 1
7325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) - 396*a**(47/2)*b**2*x**2*sqrt(1 + b*x/a)
/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4
+ 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 480*a**(47/2)*b**2*x**2/(1155*a**20*b**4 + 6930*a**19*b**5*x
+ 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b
**10*x**6) - 462*a**(45/2)*b**3*x**3*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x
**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 640*a**(
45/2)*b**3*x**3/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a
**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 480*a**(43/2)*b**4*x**4/(1155*a**20*b**4 + 69
30*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5
+ 1155*a**14*b**10*x**6) + 1848*a**(41/2)*b**5*x**5*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17
325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*
x**6) + 192*a**(41/2)*b**5*x**5/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**
7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 5544*a**(39/2)*b**6*x**6*sqrt
(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16
*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 32*a**(39/2)*b**6*x**6/(1155*a**20*b**4 + 6930*a*
*19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 11
55*a**14*b**10*x**6) + 8844*a**(37/2)*b**7*x**7*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a
**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6)
+ 8448*a**(35/2)*b**8*x**8*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 231
00*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 4840*a**(33/2)*b*
*9*x**9*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 +
17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 1540*a**(31/2)*b**10*x**10*sqrt(1 + b
*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*
x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 210*a**(29/2)*b**11*x**11*sqrt(1 + b*x/a)/(1155*a**20*b
**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b
**9*x**5 + 1155*a**14*b**10*x**6)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (56) = 112\).
time = 0.00, size = 326, normalized size = 4.53 \begin {gather*} \frac {\frac {2 b^{2} \left (\frac {1}{11} \sqrt {a+b x} \left (a+b x\right )^{5}-\frac {5}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a+\frac {10}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{2}-2 \sqrt {a+b x} \left (a+b x\right )^{2} a^{3}+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right ) a^{4}-\sqrt {a+b x} a^{5}\right )}{b^{5}}+\frac {4 a b \left (\frac {1}{9} \sqrt {a+b x} \left (a+b x\right )^{4}-\frac {4}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a+\frac {6}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{2}-\frac {4}{3} \sqrt {a+b x} \left (a+b x\right ) a^{3}+\sqrt {a+b x} a^{4}\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3}-\frac {3}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a+\sqrt {a+b x} \left (a+b x\right ) a^{2}-\sqrt {a+b x} a^{3}\right )}{b^{3}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(99*(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^2/b^3
+ 22*(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*sqr
t(b*x + a)*a^4)*a/b^3 + 5*(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)/b^3)/b

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 56, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4}+\frac {6\,a^2\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4}-\frac {2\,a\,{\left (a+b\,x\right )}^{9/2}}{3\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^(3/2),x)

[Out]

(2*(a + b*x)^(11/2))/(11*b^4) - (2*a^3*(a + b*x)^(5/2))/(5*b^4) + (6*a^2*(a + b*x)^(7/2))/(7*b^4) - (2*a*(a +
b*x)^(9/2))/(3*b^4)

________________________________________________________________________________________