3.167 \(\int x^3 \sqrt {b x^{2/3}+a x} \, dx\)

Optimal. Leaf size=371 \[ \frac {8388608 b^{12} \left (a x+b x^{2/3}\right )^{3/2}}{152108775 a^{13} x}-\frac {4194304 b^{11} \left (a x+b x^{2/3}\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac {1048576 b^{10} \left (a x+b x^{2/3}\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}-\frac {524288 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{4345965 a^{10}}+\frac {65536 b^8 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (a x+b x^{2/3}\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a} \]

[Out]

-524288/4345965*b^9*(b*x^(2/3)+a*x)^(3/2)/a^10+8388608/152108775*b^12*(b*x^(2/3)+a*x)^(3/2)/a^13/x-4194304/507
02925*b^11*(b*x^(2/3)+a*x)^(3/2)/a^12/x^(2/3)+1048576/10140585*b^10*(b*x^(2/3)+a*x)^(3/2)/a^11/x^(1/3)+65536/4
82885*b^8*x^(1/3)*(b*x^(2/3)+a*x)^(3/2)/a^9-360448/2414425*b^7*x^(2/3)*(b*x^(2/3)+a*x)^(3/2)/a^8+90112/557175*
b^6*x*(b*x^(2/3)+a*x)^(3/2)/a^7-45056/260015*b^5*x^(4/3)*(b*x^(2/3)+a*x)^(3/2)/a^6+2816/15295*b^4*x^(5/3)*(b*x
^(2/3)+a*x)^(3/2)/a^5-1408/7245*b^3*x^2*(b*x^(2/3)+a*x)^(3/2)/a^4+352/1725*b^2*x^(7/3)*(b*x^(2/3)+a*x)^(3/2)/a
^3-16/75*b*x^(8/3)*(b*x^(2/3)+a*x)^(3/2)/a^2+2/9*x^3*(b*x^(2/3)+a*x)^(3/2)/a

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Rubi [A]  time = 0.63, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ \frac {8388608 b^{12} \left (a x+b x^{2/3}\right )^{3/2}}{152108775 a^{13} x}-\frac {4194304 b^{11} \left (a x+b x^{2/3}\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac {1048576 b^{10} \left (a x+b x^{2/3}\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}-\frac {524288 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{4345965 a^{10}}+\frac {65536 b^8 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (a x+b x^{2/3}\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[b*x^(2/3) + a*x],x]

[Out]

(-524288*b^9*(b*x^(2/3) + a*x)^(3/2))/(4345965*a^10) + (8388608*b^12*(b*x^(2/3) + a*x)^(3/2))/(152108775*a^13*
x) - (4194304*b^11*(b*x^(2/3) + a*x)^(3/2))/(50702925*a^12*x^(2/3)) + (1048576*b^10*(b*x^(2/3) + a*x)^(3/2))/(
10140585*a^11*x^(1/3)) + (65536*b^8*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(482885*a^9) - (360448*b^7*x^(2/3)*(b*x^(
2/3) + a*x)^(3/2))/(2414425*a^8) + (90112*b^6*x*(b*x^(2/3) + a*x)^(3/2))/(557175*a^7) - (45056*b^5*x^(4/3)*(b*
x^(2/3) + a*x)^(3/2))/(260015*a^6) + (2816*b^4*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(15295*a^5) - (1408*b^3*x^2*(b
*x^(2/3) + a*x)^(3/2))/(7245*a^4) + (352*b^2*x^(7/3)*(b*x^(2/3) + a*x)^(3/2))/(1725*a^3) - (16*b*x^(8/3)*(b*x^
(2/3) + a*x)^(3/2))/(75*a^2) + (2*x^3*(b*x^(2/3) + a*x)^(3/2))/(9*a)

Rule 2002

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j -
1)), x] - Dist[(b*(n*p + n - j + 1))/(a*(j*p + 1)), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j,
n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int x^3 \sqrt {b x^{2/3}+a x} \, dx &=\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac {(8 b) \int x^{8/3} \sqrt {b x^{2/3}+a x} \, dx}{9 a}\\ &=-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac {\left (176 b^2\right ) \int x^{7/3} \sqrt {b x^{2/3}+a x} \, dx}{225 a^2}\\ &=\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac {\left (704 b^3\right ) \int x^2 \sqrt {b x^{2/3}+a x} \, dx}{1035 a^3}\\ &=-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac {\left (1408 b^4\right ) \int x^{5/3} \sqrt {b x^{2/3}+a x} \, dx}{2415 a^4}\\ &=\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac {\left (22528 b^5\right ) \int x^{4/3} \sqrt {b x^{2/3}+a x} \, dx}{45885 a^5}\\ &=-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac {\left (45056 b^6\right ) \int x \sqrt {b x^{2/3}+a x} \, dx}{111435 a^6}\\ &=\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac {\left (180224 b^7\right ) \int x^{2/3} \sqrt {b x^{2/3}+a x} \, dx}{557175 a^7}\\ &=-\frac {360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac {\left (360448 b^8\right ) \int \sqrt [3]{x} \sqrt {b x^{2/3}+a x} \, dx}{1448655 a^8}\\ &=\frac {65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac {\left (262144 b^9\right ) \int \sqrt {b x^{2/3}+a x} \, dx}{1448655 a^9}\\ &=-\frac {524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}+\frac {65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac {\left (524288 b^{10}\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}} \, dx}{4345965 a^{10}}\\ &=-\frac {524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}+\frac {1048576 b^{10} \left (b x^{2/3}+a x\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}+\frac {65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}-\frac {\left (2097152 b^{11}\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x^{2/3}} \, dx}{30421755 a^{11}}\\ &=-\frac {524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}-\frac {4194304 b^{11} \left (b x^{2/3}+a x\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac {1048576 b^{10} \left (b x^{2/3}+a x\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}+\frac {65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}+\frac {\left (4194304 b^{12}\right ) \int \frac {\sqrt {b x^{2/3}+a x}}{x} \, dx}{152108775 a^{12}}\\ &=-\frac {524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}+\frac {8388608 b^{12} \left (b x^{2/3}+a x\right )^{3/2}}{152108775 a^{13} x}-\frac {4194304 b^{11} \left (b x^{2/3}+a x\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac {1048576 b^{10} \left (b x^{2/3}+a x\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}+\frac {65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 181, normalized size = 0.49 \[ \frac {2 \left (a \sqrt [3]{x}+b\right ) \sqrt {a x+b x^{2/3}} \left (16900975 a^{12} x^4-16224936 a^{11} b x^{11/3}[phone] a^{10} b^2 x^{10/3}-14780480 a^9 b^3 x^3+14002560 a^8 b^4 x^{8/3}-13178880 a^7 b^5 x^{7/3}[phone] a^6 b^6 x^2-11354112 a^5 b^7 x^{5/3}[phone] a^4 b^8 x^{4/3}-9175040 a^3 b^9 x+7864320 a^2 b^{10} x^{2/3}-6291456 a b^{11} \sqrt [3]{x}+4194304 b^{12}\right )}{152108775 a^{13} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[b*x^(2/3) + a*x],x]

[Out]

(2*(b + a*x^(1/3))*Sqrt[b*x^(2/3) + a*x]*(4194304*b^12 - 6291456*a*b^11*x^(1/3) + 7864320*a^2*b^10*x^(2/3) - 9
175040*a^3*b^9*x + 10321920*a^4*b^8*x^(4/3) - 11354112*a^5*b^7*x^(5/3) + 12300288*a^6*b^6*x^2 - 13178880*a^7*b
^5*x^(7/3) + 14002560*a^8*b^4*x^(8/3) - 14780480*a^9*b^3*x^3 + 15519504*a^10*b^2*x^(10/3) - 16224936*a^11*b*x^
(11/3) + 16900975*a^12*x^4))/(152108775*a^13*x^(1/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.22, size = 396, normalized size = 1.07 \[ -\frac {8388608 \, b^{\frac {27}{2}}}{152108775 \, a^{13}} + \frac {2 \, {\left (\frac {27 \, {\left (676039 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} - 8817900 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b + 53117350 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{2} - 195695500 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{3} + 492116625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{4} - 892371480 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{5} + 1201269300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{6} - 1216870200 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{7} + 929553625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{8} - 531173500 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{9} + 223092870 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{10} - 67603900 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{11} + 16900975 \, \sqrt {a x^{\frac {1}{3}} + b} b^{12}\right )} b}{a^{12}} + \frac {13 \, {\left (1300075 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {27}{2}} - 18253053 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} b + 119041650 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b^{2} - 478056150 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{3} + 1320944625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{4} - 2657429775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{5} + 4015671660 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{6} - 4633467300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{7} + 4106936925 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{8} - 2788660875 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{9} + 1434168450 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{10} - 547591590 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{11} + 152108775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{12} - 35102025 \, \sqrt {a x^{\frac {1}{3}} + b} b^{13}\right )}}{a^{12}}\right )}}{152108775 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")

[Out]

-8388608/152108775*b^(27/2)/a^13 + 2/152108775*(27*(676039*(a*x^(1/3) + b)^(25/2) - 8817900*(a*x^(1/3) + b)^(2
3/2)*b + 53117350*(a*x^(1/3) + b)^(21/2)*b^2 - 195695500*(a*x^(1/3) + b)^(19/2)*b^3 + 492116625*(a*x^(1/3) + b
)^(17/2)*b^4 - 892371480*(a*x^(1/3) + b)^(15/2)*b^5 + 1201269300*(a*x^(1/3) + b)^(13/2)*b^6 - 1216870200*(a*x^
(1/3) + b)^(11/2)*b^7 + 929553625*(a*x^(1/3) + b)^(9/2)*b^8 - 531173500*(a*x^(1/3) + b)^(7/2)*b^9 + 223092870*
(a*x^(1/3) + b)^(5/2)*b^10 - 67603900*(a*x^(1/3) + b)^(3/2)*b^11 + 16900975*sqrt(a*x^(1/3) + b)*b^12)*b/a^12 +
13*(1300075*(a*x^(1/3) + b)^(27/2) - 18253053*(a*x^(1/3) + b)^(25/2)*b + 119041650*(a*x^(1/3) + b)^(23/2)*b^2
- 478056150*(a*x^(1/3) + b)^(21/2)*b^3 + 1320944625*(a*x^(1/3) + b)^(19/2)*b^4 - 2657429775*(a*x^(1/3) + b)^(
17/2)*b^5 + 4015671660*(a*x^(1/3) + b)^(15/2)*b^6 - 4633467300*(a*x^(1/3) + b)^(13/2)*b^7 + 4106936925*(a*x^(1
/3) + b)^(11/2)*b^8 - 2788660875*(a*x^(1/3) + b)^(9/2)*b^9 + 1434168450*(a*x^(1/3) + b)^(7/2)*b^10 - 547591590
*(a*x^(1/3) + b)^(5/2)*b^11 + 152108775*(a*x^(1/3) + b)^(3/2)*b^12 - 35102025*sqrt(a*x^(1/3) + b)*b^13)/a^12)/
a

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maple [A]  time = 0.07, size = 156, normalized size = 0.42 \[ -\frac {2 \sqrt {a x +b \,x^{\frac {2}{3}}}\, \left (a \,x^{\frac {1}{3}}+b \right ) \left (-16900975 a^{12} x^{4}[phone] a^{11} b \,x^{\frac {11}{3}}-15519504 a^{10} b^{2} x^{\frac {10}{3}}[phone] a^{9} b^{3} x^{3}-14002560 a^{8} b^{4} x^{\frac {8}{3}}[phone] a^{7} b^{5} x^{\frac {7}{3}}-12300288 a^{6} b^{6} x^{2}[phone] a^{5} b^{7} x^{\frac {5}{3}}-10321920 a^{4} b^{8} x^{\frac {4}{3}}+9175040 a^{3} b^{9} x -7864320 a^{2} b^{10} x^{\frac {2}{3}}+6291456 a \,b^{11} x^{\frac {1}{3}}-4194304 b^{12}\right )}{152108775 a^{13} x^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^(2/3)+a*x)^(1/2),x)

[Out]

-2/152108775*(b*x^(2/3)+a*x)^(1/2)*(a*x^(1/3)+b)*(16224936*x^(11/3)*a^11*b-15519504*x^(10/3)*a^10*b^2-14002560
*x^(8/3)*a^8*b^4+13178880*x^(7/3)*a^7*b^5+11354112*x^(5/3)*a^5*b^7-10321920*x^(4/3)*a^4*b^8-16900975*x^4*a^12+
14780480*x^3*a^9*b^3-7864320*x^(2/3)*a^2*b^10-12300288*x^2*a^6*b^6+6291456*x^(1/3)*a*b^11+9175040*x*a^3*b^9-41
94304*b^12)/x^(1/3)/a^13

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x + b x^{\frac {2}{3}}} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\sqrt {a\,x+b\,x^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a*x + b*x^(2/3))^(1/2),x)

[Out]

int(x^3*(a*x + b*x^(2/3))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {a x + b x^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**(2/3)+a*x)**(1/2),x)

[Out]

Integral(x**3*sqrt(a*x + b*x**(2/3)), x)

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