3.2.76 \(\int x^2 (b x^{2/3}+a x)^{3/2} \, dx\) [176]

Optimal. Leaf size=343 \[ \frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {1048576 b^{11} \left (b x^{2/3}+a x\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac {524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a} \]

[Out]

45056/557175*b^6*(b*x^(2/3)+a*x)^(5/2)/a^7-1048576/152108775*b^11*(b*x^(2/3)+a*x)^(5/2)/a^12/x^(5/3)+524288/30
421755*b^10*(b*x^(2/3)+a*x)^(5/2)/a^11/x^(4/3)-131072/4345965*b^9*(b*x^(2/3)+a*x)^(5/2)/a^10/x+65536/1448655*b
^8*(b*x^(2/3)+a*x)^(5/2)/a^9/x^(2/3)-90112/1448655*b^7*(b*x^(2/3)+a*x)^(5/2)/a^8/x^(1/3)-11264/111435*b^5*x^(1
/3)*(b*x^(2/3)+a*x)^(5/2)/a^6+5632/45885*b^4*x^(2/3)*(b*x^(2/3)+a*x)^(5/2)/a^5-352/2415*b^3*x*(b*x^(2/3)+a*x)^
(5/2)/a^4+176/1035*b^2*x^(4/3)*(b*x^(2/3)+a*x)^(5/2)/a^3-44/225*b*x^(5/3)*(b*x^(2/3)+a*x)^(5/2)/a^2+2/9*x^2*(b
*x^(2/3)+a*x)^(5/2)/a

________________________________________________________________________________________

Rubi [A]
time = 0.41, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2041, 2027, 2039} \begin {gather*} -\frac {1048576 b^{11} \left (a x+b x^{2/3}\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac {524288 b^{10} \left (a x+b x^{2/3}\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (a x+b x^{2/3}\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}+\frac {45056 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{557175 a^7}-\frac {11264 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (a x+b x^{2/3}\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(45056*b^6*(b*x^(2/3) + a*x)^(5/2))/(557175*a^7) - (1048576*b^11*(b*x^(2/3) + a*x)^(5/2))/(152108775*a^12*x^(5
/3)) + (524288*b^10*(b*x^(2/3) + a*x)^(5/2))/(30421755*a^11*x^(4/3)) - (131072*b^9*(b*x^(2/3) + a*x)^(5/2))/(4
345965*a^10*x) + (65536*b^8*(b*x^(2/3) + a*x)^(5/2))/(1448655*a^9*x^(2/3)) - (90112*b^7*(b*x^(2/3) + a*x)^(5/2
))/(1448655*a^8*x^(1/3)) - (11264*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(111435*a^6) + (5632*b^4*x^(2/3)*(b*x^(
2/3) + a*x)^(5/2))/(45885*a^5) - (352*b^3*x*(b*x^(2/3) + a*x)^(5/2))/(2415*a^4) + (176*b^2*x^(4/3)*(b*x^(2/3)
+ a*x)^(5/2))/(1035*a^3) - (44*b*x^(5/3)*(b*x^(2/3) + a*x)^(5/2))/(225*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(5/2))/
(9*a)

Rule 2027

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 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

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^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx &=\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {(22 b) \int x^{5/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{27 a}\\ &=-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (88 b^2\right ) \int x^{4/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{135 a^2}\\ &=\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (176 b^3\right ) \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx}{345 a^3}\\ &=-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (2816 b^4\right ) \int x^{2/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{7245 a^4}\\ &=\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (5632 b^5\right ) \int \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{19665 a^5}\\ &=-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (22528 b^6\right ) \int \left (b x^{2/3}+a x\right )^{3/2} \, dx}{111435 a^6}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (45056 b^7\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{334305 a^7}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (360448 b^8\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{4345965 a^8}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (65536 b^9\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{1448655 a^9}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (262144 b^{10}\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{13037895 a^{10}}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}+\frac {524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (524288 b^{11}\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{91265265 a^{11}}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {1048576 b^{11} \left (b x^{2/3}+a x\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac {524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 4.82, size = 168, normalized size = 0.49 \begin {gather*} \frac {2 \left (b+a \sqrt [3]{x}\right ) \left (b x^{2/3}+a x\right )^{3/2} \left (-524288 b^{11}+1310720 a b^{10} \sqrt [3]{x}-2293760 a^2 b^9 x^{2/3}+3440640 a^3 b^8 x-4730880 a^4 b^7 x^{4/3}+6150144 a^5 b^6 x^{5/3}-7687680 a^6 b^5 x^2+9335040 a^7 b^4 x^{7/3}-11085360 a^8 b^3 x^{8/3}[phone] a^9 b^2 x^3-14872858 a^{10} b x^{10/3}[phone] a^{11} x^{11/3}\right )}{152108775 a^{12} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x^(1/3))*(b*x^(2/3) + a*x)^(3/2)*(-524288*b^11 + 1310720*a*b^10*x^(1/3) - 2293760*a^2*b^9*x^(2/3) +
3440640*a^3*b^8*x - 4730880*a^4*b^7*x^(4/3) + 6150144*a^5*b^6*x^(5/3) - 7687680*a^6*b^5*x^2 + 9335040*a^7*b^4*
x^(7/3) - 11085360*a^8*b^3*x^(8/3) + 12932920*a^9*b^2*x^3 - 14872858*a^10*b*x^(10/3) + 16900975*a^11*x^(11/3))
)/(152108775*a^12*x)

________________________________________________________________________________________

Maple [A]
time = 0.36, size = 145, normalized size = 0.42

method	result	size
derivativedivides	\(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (16900975 a^{11} x^{\frac {11}{3}}-14872858 a^{10} b \,x^{\frac {10}{3}}[phone] a^{9} b^{2} x^{3}-11085360 a^{8} b^{3} x^{\frac {8}{3}}+9335040 a^{7} b^{4} x^{\frac {7}{3}}-7687680 a^{6} b^{5} x^{2}+6150144 a^{5} b^{6} x^{\frac {5}{3}}-4730880 a^{4} b^{7} x^{\frac {4}{3}}+3440640 a^{3} b^{8} x -2293760 a^{2} b^{9} x^{\frac {2}{3}}+1310720 a \,b^{10} x^{\frac {1}{3}}-524288 b^{11}\right )}{152108775 x \,a^{12}}\)	\(145\)
default	\(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (16900975 a^{11} x^{\frac {11}{3}}-14872858 a^{10} b \,x^{\frac {10}{3}}[phone] a^{9} b^{2} x^{3}-11085360 a^{8} b^{3} x^{\frac {8}{3}}+9335040 a^{7} b^{4} x^{\frac {7}{3}}-7687680 a^{6} b^{5} x^{2}+6150144 a^{5} b^{6} x^{\frac {5}{3}}-4730880 a^{4} b^{7} x^{\frac {4}{3}}+3440640 a^{3} b^{8} x -2293760 a^{2} b^{9} x^{\frac {2}{3}}+1310720 a \,b^{10} x^{\frac {1}{3}}-524288 b^{11}\right )}{152108775 x \,a^{12}}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^(2/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/152108775*(b*x^(2/3)+a*x)^(3/2)*(b+a*x^(1/3))*(16900975*a^11*x^(11/3)-14872858*a^10*b*x^(10/3)[phone]*a^9*
b^2*x^3-11085360*a^8*b^3*x^(8/3)+9335040*a^7*b^4*x^(7/3)-7687680*a^6*b^5*x^2+6150144*a^5*b^6*x^(5/3)-4730880*a
^4*b^7*x^(4/3)+3440640*a^3*b^8*x-2293760*a^2*b^9*x^(2/3)+1310720*a*b^10*x^(1/3)-524288*b^11)/x/a^12

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (255) = 510\).
time = 1.15, size = 770, normalized size = 2.24 \begin {gather*} \frac {2}{16900975} \, b {\left (\frac {524288 \, b^{\frac {25}{2}}}{a^{12}} + \frac {\frac {25 \, {\left (88179 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} - 1062347 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b + 5870865 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{2} - 19684665 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{3} + 44618574 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{4} - 72076158 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{5} + 85180914 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{6} - 74364290 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{7} + 47805615 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{8} - 22309287 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{9} + 7436429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{10} - 2028117 \, \sqrt {a x^{\frac {1}{3}} + b} b^{11}\right )} b}{a^{11}} + \frac {3 \, {\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 )}}{a^{11}}}{a}\right )} - \frac {2}{152108775} \, a {\left (\frac {4194304 \, b^{\frac {27}{2}}}{a^{13}} - \frac {\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}}}{a}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/16900975*b*(524288*b^(25/2)/a^12 + (25*(88179*(a*x^(1/3) + b)^(23/2) - 1062347*(a*x^(1/3) + b)^(21/2)*b + 58
70865*(a*x^(1/3) + b)^(19/2)*b^2 - 19684665*(a*x^(1/3) + b)^(17/2)*b^3 + 44618574*(a*x^(1/3) + b)^(15/2)*b^4 -
72076158*(a*x^(1/3) + b)^(13/2)*b^5 + 85180914*(a*x^(1/3) + b)^(11/2)*b^6 - 74364290*(a*x^(1/3) + b)^(9/2)*b^
7 + 47805615*(a*x^(1/3) + b)^(7/2)*b^8 - 22309287*(a*x^(1/3) + b)^(5/2)*b^9 + 7436429*(a*x^(1/3) + b)^(3/2)*b^
10 - 2028117*sqrt(a*x^(1/3) + b)*b^11)*b/a^11 + 3*(676039*(a*x^(1/3) + b)^(25/2) - 8817900*(a*x^(1/3) + b)^(23
/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)/a^11)/a)
- 2/152108775*a*(4194304*b^(27/2)/a^13 - (27*(676039*(a*x^(1/3) + b)^(25/2) - 8817900*(a*x^(1/3) + b)^(23/2)*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*(1
300075*(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 - 478
056150*(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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x + b*x^(2/3))^(3/2),x)

[Out]

int(x^2*(a*x + b*x^(2/3))^(3/2), x)

________________________________________________________________________________________