3.4.71 \(\int x^3 \sqrt [3]{a+b x} \, dx\) [371]

Optimal. Leaf size=72 \[ -\frac {3 a^3 (a+b x)^{4/3}}{4 b^4}+\frac {9 a^2 (a+b x)^{7/3}}{7 b^4}-\frac {9 a (a+b x)^{10/3}}{10 b^4}+\frac {3 (a+b x)^{13/3}}{13 b^4} \]

[Out]

-3/4*a^3*(b*x+a)^(4/3)/b^4+9/7*a^2*(b*x+a)^(7/3)/b^4-9/10*a*(b*x+a)^(10/3)/b^4+3/13*(b*x+a)^(13/3)/b^4

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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 {3 a^3 (a+b x)^{4/3}}{4 b^4}+\frac {9 a^2 (a+b x)^{7/3}}{7 b^4}+\frac {3 (a+b x)^{13/3}}{13 b^4}-\frac {9 a (a+b x)^{10/3}}{10 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(1/3),x]

[Out]

(-3*a^3*(a + b*x)^(4/3))/(4*b^4) + (9*a^2*(a + b*x)^(7/3))/(7*b^4) - (9*a*(a + b*x)^(10/3))/(10*b^4) + (3*(a +
b*x)^(13/3))/(13*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 \sqrt [3]{a+b x} \, dx &=\int \left (-\frac {a^3 \sqrt [3]{a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{4/3}}{b^3}-\frac {3 a (a+b x)^{7/3}}{b^3}+\frac {(a+b x)^{10/3}}{b^3}\right ) \, dx\\ &=-\frac {3 a^3 (a+b x)^{4/3}}{4 b^4}+\frac {9 a^2 (a+b x)^{7/3}}{7 b^4}-\frac {9 a (a+b x)^{10/3}}{10 b^4}+\frac {3 (a+b x)^{13/3}}{13 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.64 \begin {gather*} \frac {3 (a+b x)^{4/3} \left (-81 a^3+108 a^2 b x-126 a b^2 x^2+140 b^3 x^3\right )}{1820 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(1/3),x]

[Out]

(3*(a + b*x)^(4/3)*(-81*a^3 + 108*a^2*b*x - 126*a*b^2*x^2 + 140*b^3*x^3))/(1820*b^4)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(72)=144\).
time = 15.92, size = 320, normalized size = 4.44 \begin {gather*} \frac {3 a^{\frac {1}{3}} \left (81 a^{10} \left (1-\left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+27 a^9 b x \left (18-17 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+9 a^8 b^2 x^2 \left (135-119 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^7 b^3 x^3 \left (1620-1309 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+7 a^4 b^4 x^4 \left (-103 a^2+87 a b x+313 b^2 x^2\right ) \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+1215 a^6 b^4 x^4+6 a^2 b^5 x^5 \left (81 a^3+361 b^3 x^3 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^3 b^6 x^6 \left (81 a+2929 b x \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+854 a b^9 x^9 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+140 b^{10} x^{10} \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )}{1820 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)^(1/3),x]')

[Out]

3 a ^ (1 / 3) (81 a ^ 10 (1 - ((a + b x) / a) ^ (1 / 3)) + 27 a ^ 9 b x (18 - 17 ((a + b x) / a) ^ (1 / 3)) +
9 a ^ 8 b ^ 2 x ^ 2 (135 - 119 ((a + b x) / a) ^ (1 / 3)) + a ^ 7 b ^ 3 x ^ 3 (1620 - 1309 ((a + b x) / a) ^ (
1 / 3)) + 7 a ^ 4 b ^ 4 x ^ 4 (-103 a ^ 2 + 87 a b x + 313 b ^ 2 x ^ 2) ((a + b x) / a) ^ (1 / 3) + 1215 a ^ 6
b ^ 4 x ^ 4 + 6 a ^ 2 b ^ 5 x ^ 5 (81 a ^ 3 + 361 b ^ 3 x ^ 3 ((a + b x) / a) ^ (1 / 3)) + a ^ 3 b ^ 6 x ^ 6
(81 a + 2929 b x ((a + b x) / a) ^ (1 / 3)) + 854 a b ^ 9 x ^ 9 ((a + b x) / a) ^ (1 / 3) + 140 b ^ 10 x ^ 10
((a + b x) / a) ^ (1 / 3)) / (1820 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 = 0.10, size = 50, normalized size = 0.69

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (-140 b^{3} x^{3}+126 a \,b^{2} x^{2}-108 a^{2} b x +81 a^{3}\right )}{1820 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {13}{3}}}{13}-\frac {9 a \left (b x +a \right )^{\frac {10}{3}}}{10}+\frac {9 a^{2} \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a^{3} \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {13}{3}}}{13}-\frac {9 a \left (b x +a \right )^{\frac {10}{3}}}{10}+\frac {9 a^{2} \left (b x +a \right )^{\frac {7}{3}}}{7}-\frac {3 a^{3} \left (b x +a \right )^{\frac {4}{3}}}{4}}{b^{4}}\)	\(50\)
trager	\(-\frac {3 \left (-140 b^{4} x^{4}-14 a \,b^{3} x^{3}+18 a^{2} b^{2} x^{2}-27 a^{3} b x +81 a^{4}\right ) \left (b x +a \right )^{\frac {1}{3}}}{1820 b^{4}}\)	\(54\)
risch	\(-\frac {3 \left (-140 b^{4} x^{4}-14 a \,b^{3} x^{3}+18 a^{2} b^{2} x^{2}-27 a^{3} b x +81 a^{4}\right ) \left (b x +a \right )^{\frac {1}{3}}}{1820 b^{4}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/b^4*(1/13*(b*x+a)^(13/3)-3/10*a*(b*x+a)^(10/3)+3/7*a^2*(b*x+a)^(7/3)-1/4*a^3*(b*x+a)^(4/3))

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Maxima [A]
time = 0.26, size = 56, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {13}{3}}}{13 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {10}{3}} a}{10 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{2}}{7 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{3}}{4 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(1/3),x, algorithm="maxima")

[Out]

3/13*(b*x + a)^(13/3)/b^4 - 9/10*(b*x + a)^(10/3)*a/b^4 + 9/7*(b*x + a)^(7/3)*a^2/b^4 - 3/4*(b*x + a)^(4/3)*a^
3/b^4

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Fricas [A]
time = 0.31, size = 53, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (140 \, b^{4} x^{4} + 14 \, a b^{3} x^{3} - 18 \, a^{2} b^{2} x^{2} + 27 \, a^{3} b x - 81 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{1820 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(1/3),x, algorithm="fricas")

[Out]

3/1820*(140*b^4*x^4 + 14*a*b^3*x^3 - 18*a^2*b^2*x^2 + 27*a^3*b*x - 81*a^4)*(b*x + a)^(1/3)/b^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (68) = 136\).
time = 1.31, 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)**(1/3),x)

[Out]

-243*a**(73/3)*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*
b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 243*a**(73/3)/(1820*a**20
*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**
15*b**9*x**5 + 1820*a**14*b**10*x**6) - 1377*a**(70/3)*b*x*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b
**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a
**14*b**10*x**6) + 1458*a**(70/3)*b*x/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a*
*17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) - 3213*a**(67/3)*b**2*x
**2*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 +
27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 3645*a**(67/3)*b**2*x**2/(1820*a**20
*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**
15*b**9*x**5 + 1820*a**14*b**10*x**6) - 3927*a**(64/3)*b**3*x**3*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a
**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 +
1820*a**14*b**10*x**6) + 4860*a**(64/3)*b**3*x**3/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**
2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) - 2163*a**(
61/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17
*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 3645*a**(61/3)*b**4*x**4
/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4
+ 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 1827*a**(58/3)*b**5*x**5*(1 + b*x/a)**(1/3)/(1820*a**20*b*
*4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*
b**9*x**5 + 1820*a**14*b**10*x**6) + 1458*a**(58/3)*b**5*x**5/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a*
*18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6)
+ 6573*a**(55/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 +
36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 243*a**(55/3
)*b**6*x**6/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**1
6*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 8787*a**(52/3)*b**7*x**7*(1 + b*x/a)**(1/3)/(18
20*a**20*b**4 + 10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 1
0920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6) + 6498*a**(49/3)*b**8*x**8*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 +
10920*a**19*b**5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9
*x**5 + 1820*a**14*b**10*x**6) + 2562*a**(46/3)*b**9*x**9*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b*
*5*x + 27300*a**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a*
*14*b**10*x**6) + 420*a**(43/3)*b**10*x**10*(1 + b*x/a)**(1/3)/(1820*a**20*b**4 + 10920*a**19*b**5*x + 27300*a
**18*b**6*x**2 + 36400*a**17*b**7*x**3 + 27300*a**16*b**8*x**4 + 10920*a**15*b**9*x**5 + 1820*a**14*b**10*x**6
)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (56) = 112\).
time = 0.00, size = 192, normalized size = 2.67 \begin {gather*} \frac {\frac {3 b \left (\frac {1}{13} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{4}-\frac {2}{5} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3} a+\frac {6}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a^{2}-\left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{3}+\left (a+b x\right )^{\frac {1}{3}} a^{4}\right )}{b^{4}}+\frac {3 a \left (\frac {1}{10} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3}-\frac {3}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a+\frac {3}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{2}-\left (a+b x\right )^{\frac {1}{3}} 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)^(1/3),x)

[Out]

3/1820*(13*(14*(b*x + a)^(10/3) - 60*(b*x + a)^(7/3)*a + 105*(b*x + a)^(4/3)*a^2 - 140*(b*x + a)^(1/3)*a^3)*a/
b^3 + 4*(35*(b*x + a)^(13/3) - 182*(b*x + a)^(10/3)*a + 390*(b*x + a)^(7/3)*a^2 - 455*(b*x + a)^(4/3)*a^3 + 45
5*(b*x + a)^(1/3)*a^4)/b^3)/b

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Mupad [B]
time = 0.05, size = 56, normalized size = 0.78 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{13/3}}{13\,b^4}-\frac {3\,a^3\,{\left (a+b\,x\right )}^{4/3}}{4\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{7/3}}{7\,b^4}-\frac {9\,a\,{\left (a+b\,x\right )}^{10/3}}{10\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^(1/3),x)

[Out]

(3*(a + b*x)^(13/3))/(13*b^4) - (3*a^3*(a + b*x)^(4/3))/(4*b^4) + (9*a^2*(a + b*x)^(7/3))/(7*b^4) - (9*a*(a +
b*x)^(10/3))/(10*b^4)

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