3.4.87 \(\int x (a+b x)^{4/3} \, dx\) [387]

Optimal. Leaf size=34 \[ -\frac {3 a (a+b x)^{7/3}}{7 b^2}+\frac {3 (a+b x)^{10/3}}{10 b^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} \frac {3 (a+b x)^{10/3}}{10 b^2}-\frac {3 a (a+b x)^{7/3}}{7 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(4/3),x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.71 \begin {gather*} \frac {3 (a+b x)^{7/3} (-3 a+7 b x)}{70 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(4/3),x]

[Out]

(3*(a + b*x)^(7/3)*(-3*a + 7*b*x))/(70*b^2)

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Maple [A]
time = 0.11, size = 26, normalized size = 0.76

method	result	size
gosper	\(-\frac {3 \left (b x +a \right )^{\frac {7}{3}} \left (-7 b x +3 a \right )}{70 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {3 a \left (b x +a \right )^{\frac {7}{3}}}{7}}{b^{2}}\)	\(26\)
trager	\(-\frac {3 \left (-7 b^{3} x^{3}-11 a \,b^{2} x^{2}-a^{2} b x +3 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{70 b^{2}}\)	\(43\)
risch	\(-\frac {3 \left (-7 b^{3} x^{3}-11 a \,b^{2} x^{2}-a^{2} b x +3 a^{3}\right ) \left (b x +a \right )^{\frac {1}{3}}}{70 b^{2}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 26, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}}}{10 \, b^{2}} - \frac {3 \, {\left (b x + a\right )}^{\frac {7}{3}} a}{7 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.10, size = 41, normalized size = 1.21 \begin {gather*} \frac {3 \, {\left (7 \, b^{3} x^{3} + 11 \, a b^{2} x^{2} + a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{70 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/70*(7*b^3*x^3 + 11*a*b^2*x^2 + a^2*b*x - 3*a^3)*(b*x + a)^(1/3)/b^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (31) = 62\).
time = 0.19, size = 80, normalized size = 2.35 \begin {gather*} \begin {cases} - \frac {9 a^{3} \sqrt [3]{a + b x}}{70 b^{2}} + \frac {3 a^{2} x \sqrt [3]{a + b x}}{70 b} + \frac {33 a x^{2} \sqrt [3]{a + b x}}{70} + \frac {3 b x^{3} \sqrt [3]{a + b x}}{10} & \text {for}\: b \neq 0 \\\frac {a^{\frac {4}{3}} x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-9*a**3*(a + b*x)**(1/3)/(70*b**2) + 3*a**2*x*(a + b*x)**(1/3)/(70*b) + 33*a*x**2*(a + b*x)**(1/3)/
70 + 3*b*x**3*(a + b*x)**(1/3)/10, Ne(b, 0)), (a**(4/3)*x**2/2, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (26) = 52\).
time = 0.67, size = 118, normalized size = 3.47 \begin {gather*} \frac {3 \, {\left (\frac {35 \, {\left ({\left (b x + a\right )}^{\frac {4}{3}} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a^{2}}{b} + \frac {20 \, {\left (2 \, {\left (b x + a\right )}^{\frac {7}{3}} - 7 \, {\left (b x + a\right )}^{\frac {4}{3}} a + 14 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}\right )} a}{b} + \frac {14 \, {\left (b x + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2} - 140 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}}{b}\right )}}{140 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/140*(35*((b*x + a)^(4/3) - 4*(b*x + a)^(1/3)*a)*a^2/b + 20*(2*(b*x + a)^(7/3) - 7*(b*x + a)^(4/3)*a + 14*(b*
x + a)^(1/3)*a^2)*a/b + (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)/b)/b

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {30\,a\,{\left (a+b\,x\right )}^{7/3}-21\,{\left (a+b\,x\right )}^{10/3}}{70\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^(4/3),x)

[Out]

-(30*a*(a + b*x)^(7/3) - 21*(a + b*x)^(10/3))/(70*b^2)

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