3.5.13 \(\int \frac {x^3}{(a+b x)^{4/3}} \, dx\) [413]

Optimal. Leaf size=70 \[ \frac {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4} \]

[Out]

3*a^3/b^4/(b*x+a)^(1/3)+9/2*a^2*(b*x+a)^(2/3)/b^4-9/5*a*(b*x+a)^(5/3)/b^4+3/8*(b*x+a)^(8/3)/b^4

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Rubi [A]
time = 0.01, antiderivative size = 70, 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 {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x)^(4/3),x]

[Out]

(3*a^3)/(b^4*(a + b*x)^(1/3)) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) - (9*a*(a + b*x)^(5/3))/(5*b^4) + (3*(a + b*x)
^(8/3))/(8*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)^{4/3}} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^{4/3}}+\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {(a+b x)^{5/3}}{b^3}\right ) \, dx\\ &=\frac {3 a^3}{b^4 \sqrt [3]{a+b x}}+\frac {9 a^2 (a+b x)^{2/3}}{2 b^4}-\frac {9 a (a+b x)^{5/3}}{5 b^4}+\frac {3 (a+b x)^{8/3}}{8 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.66 \begin {gather*} \frac {3 \left (81 a^3+27 a^2 b x-9 a b^2 x^2+5 b^3 x^3\right )}{40 b^4 \sqrt [3]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x)^(4/3),x]

[Out]

(3*(81*a^3 + 27*a^2*b*x - 9*a*b^2*x^2 + 5*b^3*x^3))/(40*b^4*(a + b*x)^(1/3))

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(295\) vs. \(2(70)=140\).
time = 14.77, size = 277, normalized size = 3.96 \begin {gather*} \frac {3 a^{\frac {2}{3}} \left (81 a^8 \left (-1+\left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+54 a^7 b x \left (-9+8 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+9 a^6 b^2 x^2 \left (-135+104 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+20 a^5 b^3 x^3 \left (-81+52 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+5 b^4 x^4 \left (-243 a^4+b^4 x^4 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}\right )+610 a^4 b^4 x^4 \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}+16 a b^5 x^5 \left (11 a^2+2 a b x+b^2 x^2\right ) \left (\frac {a+b x}{a}\right )^{\frac {2}{3}}-486 a^3 b^5 x^5-81 a^2 b^6 x^6\right )}{40 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)^(4/3),x]')

[Out]

3 a ^ (2 / 3) (81 a ^ 8 (-1 + ((a + b x) / a) ^ (2 / 3)) + 54 a ^ 7 b x (-9 + 8 ((a + b x) / a) ^ (2 / 3)) + 9
a ^ 6 b ^ 2 x ^ 2 (-135 + 104 ((a + b x) / a) ^ (2 / 3)) + 20 a ^ 5 b ^ 3 x ^ 3 (-81 + 52 ((a + b x) / a) ^ (
2 / 3)) + 5 b ^ 4 x ^ 4 (-243 a ^ 4 + b ^ 4 x ^ 4 ((a + b x) / a) ^ (2 / 3)) + 610 a ^ 4 b ^ 4 x ^ 4 ((a + b x
) / a) ^ (2 / 3) + 16 a b ^ 5 x ^ 5 (11 a ^ 2 + 2 a b x + b ^ 2 x ^ 2) ((a + b x) / a) ^ (2 / 3) - 486 a ^ 3 b
^ 5 x ^ 5 - 81 a ^ 2 b ^ 6 x ^ 6) / (40 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))

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Maple [A]
time = 0.12, size = 49, normalized size = 0.70

method	result	size
gosper	\(\frac {\frac {3}{8} b^{3} x^{3}-\frac {27}{40} a \,b^{2} x^{2}+\frac {81}{40} a^{2} b x +\frac {243}{40} a^{3}}{\left (b x +a \right )^{\frac {1}{3}} b^{4}}\)	\(43\)
trager	\(\frac {\frac {3}{8} b^{3} x^{3}-\frac {27}{40} a \,b^{2} x^{2}+\frac {81}{40} a^{2} b x +\frac {243}{40} a^{3}}{\left (b x +a \right )^{\frac {1}{3}} b^{4}}\)	\(43\)
risch	\(\frac {3 \left (5 x^{2} b^{2}-14 a b x +41 a^{2}\right ) \left (b x +a \right )^{\frac {2}{3}}}{40 b^{4}}+\frac {3 a^{3}}{b^{4} \left (b x +a \right )^{\frac {1}{3}}}\)	\(48\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {9 a \left (b x +a \right )^{\frac {5}{3}}}{5}+\frac {9 a^{2} \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a^{3}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{4}}\)	\(49\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {8}{3}}}{8}-\frac {9 a \left (b x +a \right )^{\frac {5}{3}}}{5}+\frac {9 a^{2} \left (b x +a \right )^{\frac {2}{3}}}{2}+\frac {3 a^{3}}{\left (b x +a \right )^{\frac {1}{3}}}}{b^{4}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x+a)^(4/3),x,method=_RETURNVERBOSE)

[Out]

3/b^4*(1/8*(b*x+a)^(8/3)-3/5*a*(b*x+a)^(5/3)+3/2*a^2*(b*x+a)^(2/3)+a^3/(b*x+a)^(1/3))

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Maxima [A]
time = 0.27, size = 56, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {8}{3}}}{8 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {5}{3}} a}{5 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{2}}{2 \, b^{4}} + \frac {3 \, a^{3}}{{\left (b x + a\right )}^{\frac {1}{3}} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^(4/3),x, algorithm="maxima")

[Out]

3/8*(b*x + a)^(8/3)/b^4 - 9/5*(b*x + a)^(5/3)*a/b^4 + 9/2*(b*x + a)^(2/3)*a^2/b^4 + 3*a^3/((b*x + a)^(1/3)*b^4
)

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Fricas [A]
time = 0.30, size = 52, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (5 \, b^{3} x^{3} - 9 \, a b^{2} x^{2} + 27 \, a^{2} b x + 81 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{40 \, {\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)^(4/3),x, algorithm="fricas")

[Out]

3/40*(5*b^3*x^3 - 9*a*b^2*x^2 + 27*a^2*b*x + 81*a^3)*(b*x + a)^(2/3)/(b^5*x + a*b^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1538 vs. \(2 (66) = 132\).
time = 1.38, size = 1538, normalized size = 21.97

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x+a)**(4/3),x)

[Out]

243*a**(68/3)*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3
+ 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 243*a**(68/3)/(40*a**20*b**4 + 240*a**19
*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**
10*x**6) + 1296*a**(65/3)*b*x*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800
*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(65/3)*b*x/(40*a
**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**
9*x**5 + 40*a**14*b**10*x**6) + 2808*a**(62/3)*b**2*x**2*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x
+ 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6)
- 3645*a**(62/3)*b**2*x**2/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 60
0*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 3120*a**(59/3)*b**3*x**3*(1 + b*x/a)**(2/3)/(
40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15
*b**9*x**5 + 40*a**14*b**10*x**6) - 4860*a**(59/3)*b**3*x**3/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**
6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 1830*a**(56/
3)*b**4*x**4*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3
+ 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 3645*a**(56/3)*b**4*x**4/(40*a**20*b**4 +
240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6) + 528*a**(53/3)*b**5*x**5*(1 + b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*
b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) - 1458*a**(
53/3)*b**5*x**5/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8
*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 96*a**(50/3)*b**6*x**6*(1 + b*x/a)**(2/3)/(40*a**20*b**4
+ 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 4
0*a**14*b**10*x**6) - 243*a**(50/3)*b**6*x**6/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a*
*17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 48*a**(47/3)*b**7*x**7*(1 +
b*x/a)**(2/3)/(40*a**20*b**4 + 240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*
x**4 + 240*a**15*b**9*x**5 + 40*a**14*b**10*x**6) + 15*a**(44/3)*b**8*x**8*(1 + b*x/a)**(2/3)/(40*a**20*b**4 +
240*a**19*b**5*x + 600*a**18*b**6*x**2 + 800*a**17*b**7*x**3 + 600*a**16*b**8*x**4 + 240*a**15*b**9*x**5 + 40
*a**14*b**10*x**6)

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Giac [A]
time = 0.00, size = 97, normalized size = 1.39 \begin {gather*} 3 \left (\frac {\frac {1}{8} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right )^{2} b^{28}-\frac {3}{5} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} \left (a+b x\right ) a b^{28}+\frac {3}{2} \left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2} a^{2} b^{28}}{b^{32}}+\frac {a^{3}}{b^{4} \left (a+b x\right )^{\frac {1}{3}}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^(4/3),x)

[Out]

3*a^3/((b*x + a)^(1/3)*b^4) + 3/40*(5*(b*x + a)^(8/3)*b^28 - 24*(b*x + a)^(5/3)*a*b^28 + 60*(b*x + a)^(2/3)*a^
2*b^28)/b^32

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Mupad [B]
time = 0.05, size = 56, normalized size = 0.80 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{8/3}}{8\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{2/3}}{2\,b^4}+\frac {3\,a^3}{b^4\,{\left (a+b\,x\right )}^{1/3}}-\frac {9\,a\,{\left (a+b\,x\right )}^{5/3}}{5\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x)^(4/3),x)

[Out]

(3*(a + b*x)^(8/3))/(8*b^4) + (9*a^2*(a + b*x)^(2/3))/(2*b^4) + (3*a^3)/(b^4*(a + b*x)^(1/3)) - (9*a*(a + b*x)
^(5/3))/(5*b^4)

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