3.4.44 \(\int \frac {x^3}{(a+b x)^{3/2}} \, dx\) [344]

Optimal. Leaf size=66 \[ \frac {2 a^3}{b^4 \sqrt {a+b x}}+\frac {6 a^2 \sqrt {a+b x}}{b^4}-\frac {2 a (a+b x)^{3/2}}{b^4}+\frac {2 (a+b x)^{5/2}}{5 b^4} \]

[Out]

-2*a*(b*x+a)^(3/2)/b^4+2/5*(b*x+a)^(5/2)/b^4+2*a^3/b^4/(b*x+a)^(1/2)+6*a^2*(b*x+a)^(1/2)/b^4

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 66, 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}{b^4 \sqrt {a+b x}}+\frac {6 a^2 \sqrt {a+b x}}{b^4}-\frac {2 a (a+b x)^{3/2}}{b^4}+\frac {2 (a+b x)^{5/2}}{5 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 45, normalized size = 0.68 \begin {gather*} \frac {2 \left (16 a^3+8 a^2 b x-2 a b^2 x^2+b^3 x^3\right )}{5 b^4 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x)^(3/2),x]

[Out]

(2*(16*a^3 + 8*a^2*b*x - 2*a*b^2*x^2 + b^3*x^3))/(5*b^4*Sqrt[a + b*x])

________________________________________________________________________________________

Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(302\) vs. \(2(66)=132\).
time = 14.56, size = 280, normalized size = 4.24 \begin {gather*} \frac {2 \sqrt {a} \left (16 a^8 \left (-1+\sqrt {\frac {a+b x}{a}}\right )+8 a^7 b x \left (-12+11 \sqrt {\frac {a+b x}{a}}\right )+6 a^6 b^2 x^2 \left (-40+33 \sqrt {\frac {a+b x}{a}}\right )+a^5 b^3 x^3 \left (-320+231 \sqrt {\frac {a+b x}{a}}\right )+5 a^4 b^4 x^4 \left (-48+29 \sqrt {\frac {a+b x}{a}}\right )+b^5 x^5 \left (-96 a^3+46 a^3 \sqrt {\frac {a+b x}{a}}-16 a^2 b x+8 a^2 b x \sqrt {\frac {a+b x}{a}}+3 a b^2 x^2 \sqrt {\frac {a+b x}{a}}+b^3 x^3 \sqrt {\frac {a+b x}{a}}\right )\right )}{5 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 ^ 8 (-1 + Sqrt[(a + b x) / a]) + 8 a ^ 7 b x (-12 + 11 Sqrt[(a + b x) / a]) + 6 a ^ 6 b ^ 2 x
^ 2 (-40 + 33 Sqrt[(a + b x) / a]) + a ^ 5 b ^ 3 x ^ 3 (-320 + 231 Sqrt[(a + b x) / a]) + 5 a ^ 4 b ^ 4 x ^ 4
(-48 + 29 Sqrt[(a + b x) / a]) + b ^ 5 x ^ 5 (-96 a ^ 3 + 46 a ^ 3 Sqrt[(a + b x) / a] - 16 a ^ 2 b x + 8 a ^
2 b x Sqrt[(a + b x) / a] + 3 a b ^ 2 x ^ 2 Sqrt[(a + b x) / a] + b ^ 3 x ^ 3 Sqrt[(a + b x) / a])) / (5 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 = 0.11, size = 49, normalized size = 0.74

method	result	size
gosper	\(\frac {\frac {2}{5} b^{3} x^{3}-\frac {4}{5} a \,b^{2} x^{2}+\frac {16}{5} a^{2} b x +\frac {32}{5} a^{3}}{b^{4} \sqrt {b x +a}}\)	\(42\)
trager	\(\frac {\frac {2}{5} b^{3} x^{3}-\frac {4}{5} a \,b^{2} x^{2}+\frac {16}{5} a^{2} b x +\frac {32}{5} a^{3}}{b^{4} \sqrt {b x +a}}\)	\(42\)
risch	\(\frac {2 \left (x^{2} b^{2}-3 a b x +11 a^{2}\right ) \sqrt {b x +a}}{5 b^{4}}+\frac {2 a^{3}}{b^{4} \sqrt {b x +a}}\)	\(47\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-2 a \left (b x +a \right )^{\frac {3}{2}}+6 a^{2} \sqrt {b x +a}+\frac {2 a^{3}}{\sqrt {b x +a}}}{b^{4}}\)	\(49\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-2 a \left (b x +a \right )^{\frac {3}{2}}+6 a^{2} \sqrt {b x +a}+\frac {2 a^{3}}{\sqrt {b x +a}}}{b^{4}}\)	\(49\)

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*(1/5*(b*x+a)^(5/2)-a*(b*x+a)^(3/2)+3*a^2*(b*x+a)^(1/2)+a^3/(b*x+a)^(1/2))

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 56, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{b^{4}} + \frac {6 \, \sqrt {b x + a} a^{2}}{b^{4}} + \frac {2 \, a^{3}}{\sqrt {b x + a} 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]

2/5*(b*x + a)^(5/2)/b^4 - 2*(b*x + a)^(3/2)*a/b^4 + 6*sqrt(b*x + a)*a^2/b^4 + 2*a^3/(sqrt(b*x + a)*b^4)

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 51, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b^{3} x^{3} - 2 \, a b^{2} x^{2} + 8 \, a^{2} b x + 16 \, a^{3}\right )} \sqrt {b x + a}}{5 \, {\left (b^{5} x + a b^{4}\right )}} \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]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1538 vs. \(2 (63) = 126\).
time = 1.34, size = 1538, normalized size = 23.30

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**(45/2)*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a
**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 32*a**(45/2)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*
a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 176*a*
*(43/2)*b*x*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a*
*16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 192*a**(43/2)*b*x/(5*a**20*b**4 + 30*a**19*b**5*x +
75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 39
6*a**(41/2)*b**2*x**2*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x*
*3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 480*a**(41/2)*b**2*x**2/(5*a**20*b**4 + 3
0*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*
b**10*x**6) + 462*a**(39/2)*b**3*x**3*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 1
00*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 640*a**(39/2)*b**3*x**3/(
5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9
*x**5 + 5*a**14*b**10*x**6) + 290*a**(37/2)*b**4*x**4*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**
18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 480*a**(3
7/2)*b**4*x**4/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4
+ 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 92*a**(35/2)*b**5*x**5*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*
b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x*
*6) - 192*a**(35/2)*b**5*x**5/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*
a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 16*a**(33/2)*b**6*x**6*sqrt(1 + b*x/a)/(5*a**20*b
**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5
*a**14*b**10*x**6) - 32*a**(33/2)*b**6*x**6/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b
**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 6*a**(31/2)*b**7*x**7*sqrt(1 + b*x/
a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*
b**9*x**5 + 5*a**14*b**10*x**6) + 2*a**(29/2)*b**8*x**8*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a
**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 86, normalized size = 1.30 \begin {gather*} 2 \left (\frac {\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2} b^{16}-\sqrt {a+b x} \left (a+b x\right ) a b^{16}+3 \sqrt {a+b x} a^{2} b^{16}}{b^{20}}+\frac {a^{3}}{b^{4} \sqrt {a+b x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^3/(sqrt(b*x + a)*b^4) + 2/5*((b*x + a)^(5/2)*b^16 - 5*(b*x + a)^(3/2)*a*b^16 + 15*sqrt(b*x + a)*a^2*b^16)/
b^20

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 56, normalized size = 0.85 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4}+\frac {6\,a^2\,\sqrt {a+b\,x}}{b^4}+\frac {2\,a^3}{b^4\,\sqrt {a+b\,x}}-\frac {2\,a\,{\left (a+b\,x\right )}^{3/2}}{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)^(5/2))/(5*b^4) + (6*a^2*(a + b*x)^(1/2))/b^4 + (2*a^3)/(b^4*(a + b*x)^(1/2)) - (2*a*(a + b*x)^(3/
2))/b^4

________________________________________________________________________________________