3.4.85 \(\int x^3 (a+b x)^{4/3} \, dx\) [385]

Optimal. Leaf size=72 \[ -\frac {3 a^3 (a+b x)^{7/3}}{7 b^4}+\frac {9 a^2 (a+b x)^{10/3}}{10 b^4}-\frac {9 a (a+b x)^{13/3}}{13 b^4}+\frac {3 (a+b x)^{16/3}}{16 b^4} \]

[Out]

-3/7*a^3*(b*x+a)^(7/3)/b^4+9/10*a^2*(b*x+a)^(10/3)/b^4-9/13*a*(b*x+a)^(13/3)/b^4+3/16*(b*x+a)^(16/3)/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)^{7/3}}{7 b^4}+\frac {9 a^2 (a+b x)^{10/3}}{10 b^4}+\frac {3 (a+b x)^{16/3}}{16 b^4}-\frac {9 a (a+b x)^{13/3}}{13 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(4/3),x]

[Out]

(-3*a^3*(a + b*x)^(7/3))/(7*b^4) + (9*a^2*(a + b*x)^(10/3))/(10*b^4) - (9*a*(a + b*x)^(13/3))/(13*b^4) + (3*(a
+ b*x)^(16/3))/(16*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)^{4/3} \, dx &=\int \left (-\frac {a^3 (a+b x)^{4/3}}{b^3}+\frac {3 a^2 (a+b x)^{7/3}}{b^3}-\frac {3 a (a+b x)^{10/3}}{b^3}+\frac {(a+b x)^{13/3}}{b^3}\right ) \, dx\\ &=-\frac {3 a^3 (a+b x)^{7/3}}{7 b^4}+\frac {9 a^2 (a+b x)^{10/3}}{10 b^4}-\frac {9 a (a+b x)^{13/3}}{13 b^4}+\frac {3 (a+b x)^{16/3}}{16 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)^{7/3} \left (-81 a^3+189 a^2 b x-315 a b^2 x^2+455 b^3 x^3\right )}{7280 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3)*(-81*a^3 + 189*a^2*b*x - 315*a*b^2*x^2 + 455*b^3*x^3))/(7280*b^4)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(372\) vs. \(2(72)=144\).
time = 17.18, size = 348, normalized size = 4.83 \begin {gather*} \frac {3 a^{\frac {1}{3}} \left (81 a^{11} \left (1-\left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+27 a^{10} b x \left (18-17 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+9 a^9 b^2 x^2 \left (135-119 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^8 b^3 x^3 \left (1620-1309 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+14 a^5 b^4 x^4 \left (-19 a^2+271 a b x+839 b^2 x^2\right ) \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+1215 a^7 b^4 x^4+2 a^4 b^5 x^5 \left (243 a^2+9427 b^2 x^2 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^3 b^6 x^6 \left (81 a^2+18091 b^2 x^2 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+10409 a^2 b^9 x^9 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+3325 a b^{10} x^{10} \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+455 b^{11} x^{11} \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )}{7280 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 ^ (1 / 3) (81 a ^ 11 (1 - ((a + b x) / a) ^ (1 / 3)) + 27 a ^ 10 b x (18 - 17 ((a + b x) / a) ^ (1 / 3)) +
9 a ^ 9 b ^ 2 x ^ 2 (135 - 119 ((a + b x) / a) ^ (1 / 3)) + a ^ 8 b ^ 3 x ^ 3 (1620 - 1309 ((a + b x) / a) ^
(1 / 3)) + 14 a ^ 5 b ^ 4 x ^ 4 (-19 a ^ 2 + 271 a b x + 839 b ^ 2 x ^ 2) ((a + b x) / a) ^ (1 / 3) + 1215 a ^
7 b ^ 4 x ^ 4 + 2 a ^ 4 b ^ 5 x ^ 5 (243 a ^ 2 + 9427 b ^ 2 x ^ 2 ((a + b x) / a) ^ (1 / 3)) + a ^ 3 b ^ 6 x
^ 6 (81 a ^ 2 + 18091 b ^ 2 x ^ 2 ((a + b x) / a) ^ (1 / 3)) + 10409 a ^ 2 b ^ 9 x ^ 9 ((a + b x) / a) ^ (1 /
3) + 3325 a b ^ 10 x ^ 10 ((a + b x) / a) ^ (1 / 3) + 455 b ^ 11 x ^ 11 ((a + b x) / a) ^ (1 / 3)) / (7280 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 = 0.11, size = 50, normalized size = 0.69

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (-455 b^{3} x^{3}+315 a \,b^{2} x^{2}-189 a^{2} b x +81 a^{3}\right )}{7280 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {16}{3}}}{16}-\frac {9 a \left (b x +a \right )^{\frac {13}{3}}}{13}+\frac {9 a^{2} \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a^{3} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {16}{3}}}{16}-\frac {9 a \left (b x +a \right )^{\frac {13}{3}}}{13}+\frac {9 a^{2} \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a^{3} \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{4}}\)	\(50\)
trager	\(-\frac {3 \left (-455 b^{5} x^{5}-595 a \,b^{4} x^{4}-14 a^{2} b^{3} x^{3}+18 a^{3} b^{2} x^{2}-27 a^{4} b x +81 a^{5}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7280 b^{4}}\)	\(65\)
risch	\(-\frac {3 \left (-455 b^{5} x^{5}-595 a \,b^{4} x^{4}-14 a^{2} b^{3} x^{3}+18 a^{3} b^{2} x^{2}-27 a^{4} b x +81 a^{5}\right ) \left (b x +a \right )^{\frac {1}{3}}}{7280 b^{4}}\)	\(65\)

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*(1/16*(b*x+a)^(16/3)-3/13*a*(b*x+a)^(13/3)+3/10*a^2*(b*x+a)^(10/3)-1/7*a^3*(b*x+a)^(7/3))

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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 {16}{3}}}{16 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {13}{3}} a}{13 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {10}{3}} a^{2}}{10 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a^{3}}{7 \, 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]

3/16*(b*x + a)^(16/3)/b^4 - 9/13*(b*x + a)^(13/3)*a/b^4 + 9/10*(b*x + a)^(10/3)*a^2/b^4 - 3/7*(b*x + a)^(7/3)*
a^3/b^4

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Fricas [A]
time = 0.30, size = 64, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (455 \, b^{5} x^{5} + 595 \, a b^{4} x^{4} + 14 \, a^{2} b^{3} x^{3} - 18 \, a^{3} b^{2} x^{2} + 27 \, a^{4} b x - 81 \, a^{5}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{7280 \, 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="fricas")

[Out]

3/7280*(455*b^5*x^5 + 595*a*b^4*x^4 + 14*a^2*b^3*x^3 - 18*a^3*b^2*x^2 + 27*a^4*b*x - 81*a^5)*(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. 1844 vs. \(2 (68) = 136\).
time = 1.48, size = 1844, normalized size = 25.61

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**(76/3)*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**1
7*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 243*a**(76/3)/(7280*a*
*20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 436
80*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) - 1377*a**(73/3)*b*x*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a
**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5
+ 7280*a**14*b**10*x**6) + 1458*a**(73/3)*b*x/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2
+ 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) - 3213*a**(
70/3)*b**2*x**2*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**
17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 3645*a**(70/3)*b**2*x
**2/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**
8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) - 3927*a**(67/3)*b**3*x**3*(1 + b*x/a)**(1/3)/(7280*a*
*20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 436
80*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 4860*a**(67/3)*b**3*x**3/(7280*a**20*b**4 + 43680*a**19*b**5*x +
109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14
*b**10*x**6) - 798*a**(64/3)*b**4*x**4*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18
*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6)
+ 3645*a**(64/3)*b**4*x**4/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*
x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 11382*a**(61/3)*b**5*x**5*(1
+ b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 1092
00*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 1458*a**(61/3)*b**5*x**5/(7280*a**20*b**
4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**1
5*b**9*x**5 + 7280*a**14*b**10*x**6) + 35238*a**(58/3)*b**6*x**6*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a
**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5
+ 7280*a**14*b**10*x**6) + 243*a**(58/3)*b**6*x**6/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*
x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 5656
2*a**(55/3)*b**7*x**7*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 1456
00*a**17*b**7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 54273*a**(52/3)
*b**8*x**8*(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b*
*7*x**3 + 109200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 31227*a**(49/3)*b**9*x**9*
(1 + b*x/a)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 1
09200*a**16*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 9975*a**(46/3)*b**10*x**10*(1 + b*x/a
)**(1/3)/(7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**1
6*b**8*x**4 + 43680*a**15*b**9*x**5 + 7280*a**14*b**10*x**6) + 1365*a**(43/3)*b**11*x**11*(1 + b*x/a)**(1/3)/(
7280*a**20*b**4 + 43680*a**19*b**5*x + 109200*a**18*b**6*x**2 + 145600*a**17*b**7*x**3 + 109200*a**16*b**8*x**
4 + 43680*a**15*b**9*x**5 + 7280*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. 193 vs. \(2 (56) = 112\).
time = 0.00, size = 325, normalized size = 4.51 \begin {gather*} \frac {\frac {3 b^{2} \left (\frac {1}{16} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{5}-\frac {5}{13} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{4} a+\left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3} a^{2}-\frac {10}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a^{3}+\frac {5}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{4}-\left (a+b x\right )^{\frac {1}{3}} a^{5}\right )}{b^{5}}+\frac {6 a 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^{2} \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)^(4/3),x)

[Out]

3/7280*(52*(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^
2/b^3 + 32*(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 +
455*(b*x + a)^(1/3)*a^4)*a/b^3 + 5*(91*(b*x + a)^(16/3) - 560*(b*x + a)^(13/3)*a + 1456*(b*x + a)^(10/3)*a^2
- 2080*(b*x + a)^(7/3)*a^3 + 1820*(b*x + a)^(4/3)*a^4 - 1456*(b*x + a)^(1/3)*a^5)/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 )}^{16/3}}{16\,b^4}-\frac {3\,a^3\,{\left (a+b\,x\right )}^{7/3}}{7\,b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{10/3}}{10\,b^4}-\frac {9\,a\,{\left (a+b\,x\right )}^{13/3}}{13\,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)^(16/3))/(16*b^4) - (3*a^3*(a + b*x)^(7/3))/(7*b^4) + (9*a^2*(a + b*x)^(10/3))/(10*b^4) - (9*a*(a
+ b*x)^(13/3))/(13*b^4)

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