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

Optimal. Leaf size=255 \[ \frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac {256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \]

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Rubi [A]  time = 0.42, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2016, 2002, 2014} \begin {gather*} \frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac {256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-256*b^3*(b*x^(2/3) + a*x)^(5/2))/(1615*a^4) + (65536*b^8*(b*x^(2/3) + a*x)^(5/2))/(4849845*a^9*x^(5/3)) - (3
2768*b^7*(b*x^(2/3) + a*x)^(5/2))/(969969*a^8*x^(4/3)) + (8192*b^6*(b*x^(2/3) + a*x)^(5/2))/(138567*a^7*x) - (
4096*b^5*(b*x^(2/3) + a*x)^(5/2))/(46189*a^6*x^(2/3)) + (512*b^4*(b*x^(2/3) + a*x)^(5/2))/(4199*a^5*x^(1/3)) +
(64*b^2*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(323*a^3) - (32*b*x^(2/3)*(b*x^(2/3) + a*x)^(5/2))/(133*a^2) + (2*x*
(b*x^(2/3) + a*x)^(5/2))/(7*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 \left (b x^{2/3}+a x\right )^{3/2} \, dx &=\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {(16 b) \int x^{2/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{21 a}\\ &=-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (32 b^2\right ) \int \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{57 a^2}\\ &=\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (128 b^3\right ) \int \left (b x^{2/3}+a x\right )^{3/2} \, dx}{323 a^3}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (256 b^4\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{969 a^4}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (2048 b^5\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{12597 a^5}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (4096 b^6\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{46189 a^6}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (16384 b^7\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{415701 a^7}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}-\frac {32768 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (32768 b^8\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{2909907 a^8}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 135, normalized size = 0.53 \begin {gather*} \frac {2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt {a x+b x^{2/3}} \left (692835 a^8 x^{8/3}-583440 a^7 b x^{7/3}+480480 a^6 b^2 x^2-384384 a^5 b^3 x^{5/3}+295680 a^4 b^4 x^{4/3}-215040 a^3 b^5 x+143360 a^2 b^6 x^{2/3}-81920 a b^7 \sqrt [3]{x}+32768 b^8\right )}{4849845 a^9 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*(32768*b^8 - 81920*a*b^7*x^(1/3) + 143360*a^2*b^6*x^(2/3) - 215040*
a^3*b^5*x + 295680*a^4*b^4*x^(4/3) - 384384*a^5*b^3*x^(5/3) + 480480*a^6*b^2*x^2 - 583440*a^7*b*x^(7/3) + 6928
35*a^8*x^(8/3)))/(4849845*a^9*x^(1/3))

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IntegrateAlgebraic [A]  time = 4.50, size = 161, normalized size = 0.63 \begin {gather*} \frac {2 \left (x^{2/3} \left (a \sqrt [3]{x}+b\right )\right )^{3/2} \left (692835 a^{10} x^{10/3}+802230 a^9 b x^3+6435 a^8 b^2 x^{8/3}-6864 a^7 b^3 x^{7/3}+7392 a^6 b^4 x^2-8064 a^5 b^5 x^{5/3}+8960 a^4 b^6 x^{4/3}-10240 a^3 b^7 x+12288 a^2 b^8 x^{2/3}-16384 a b^9 \sqrt [3]{x}+32768 b^{10}\right )}{4849845 a^9 x \left (a \sqrt [3]{x}+b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((b + a*x^(1/3))*x^(2/3))^(3/2)*(32768*b^10 - 16384*a*b^9*x^(1/3) + 12288*a^2*b^8*x^(2/3) - 10240*a^3*b^7*x
+ 8960*a^4*b^6*x^(4/3) - 8064*a^5*b^5*x^(5/3) + 7392*a^6*b^4*x^2 - 6864*a^7*b^3*x^(7/3) + 6435*a^8*b^2*x^(8/3
) + 802230*a^9*b*x^3 + 692835*a^10*x^(10/3)))/(4849845*a^9*(b + a*x^(1/3))*x)

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fricas [F(-1)]  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*(b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.30, size = 602, normalized size = 2.36 \begin {gather*} -\frac {2}{692835} \, b {\left (\frac {32768 \, b^{\frac {19}{2}}}{a^{9}} - \frac {\frac {19 \, {\left (6435 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} - 58344 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b + 235620 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{2} - 556920 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{3} + 850850 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{4} - 875160 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{5} + 612612 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{6} - 291720 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{7} + 109395 \, \sqrt {a x^{\frac {1}{3}} + b} b^{8}\right )} b}{a^{8}} + \frac {9 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )}}{a^{8}}}{a}\right )} + \frac {2}{1616615} \, a {\left (\frac {65536 \, b^{\frac {21}{2}}}{a^{10}} + \frac {\frac {21 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )} b}{a^{9}} + \frac {5 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{a^{9}}}{a}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/692835*b*(32768*b^(19/2)/a^9 - (19*(6435*(a*x^(1/3) + b)^(17/2) - 58344*(a*x^(1/3) + b)^(15/2)*b + 235620*(
a*x^(1/3) + b)^(13/2)*b^2 - 556920*(a*x^(1/3) + b)^(11/2)*b^3 + 850850*(a*x^(1/3) + b)^(9/2)*b^4 - 875160*(a*x
^(1/3) + b)^(7/2)*b^5 + 612612*(a*x^(1/3) + b)^(5/2)*b^6 - 291720*(a*x^(1/3) + b)^(3/2)*b^7 + 109395*sqrt(a*x^
(1/3) + b)*b^8)*b/a^8 + 9*(12155*(a*x^(1/3) + b)^(19/2) - 122265*(a*x^(1/3) + b)^(17/2)*b + 554268*(a*x^(1/3)
+ b)^(15/2)*b^2 - 1492260*(a*x^(1/3) + b)^(13/2)*b^3 + 2645370*(a*x^(1/3) + b)^(11/2)*b^4 - 3233230*(a*x^(1/3)
+ b)^(9/2)*b^5 + 2771340*(a*x^(1/3) + b)^(7/2)*b^6 - 1662804*(a*x^(1/3) + b)^(5/2)*b^7 + 692835*(a*x^(1/3) +
b)^(3/2)*b^8 - 230945*sqrt(a*x^(1/3) + b)*b^9)/a^8)/a) + 2/1616615*a*(65536*b^(21/2)/a^10 + (21*(12155*(a*x^(1
/3) + b)^(19/2) - 122265*(a*x^(1/3) + b)^(17/2)*b + 554268*(a*x^(1/3) + b)^(15/2)*b^2 - 1492260*(a*x^(1/3) + b
)^(13/2)*b^3 + 2645370*(a*x^(1/3) + b)^(11/2)*b^4 - 3233230*(a*x^(1/3) + b)^(9/2)*b^5 + 2771340*(a*x^(1/3) + b
)^(7/2)*b^6 - 1662804*(a*x^(1/3) + b)^(5/2)*b^7 + 692835*(a*x^(1/3) + b)^(3/2)*b^8 - 230945*sqrt(a*x^(1/3) + b
)*b^9)*b/a^9 + 5*(46189*(a*x^(1/3) + b)^(21/2) - 510510*(a*x^(1/3) + b)^(19/2)*b + 2567565*(a*x^(1/3) + b)^(17
/2)*b^2 - 7759752*(a*x^(1/3) + b)^(15/2)*b^3 + 15668730*(a*x^(1/3) + b)^(13/2)*b^4 - 22221108*(a*x^(1/3) + b)^
(11/2)*b^5 + 22632610*(a*x^(1/3) + b)^(9/2)*b^6 - 16628040*(a*x^(1/3) + b)^(7/2)*b^7 + 8729721*(a*x^(1/3) + b)
^(5/2)*b^8 - 3233230*(a*x^(1/3) + b)^(3/2)*b^9 + 969969*sqrt(a*x^(1/3) + b)*b^10)/a^9)/a)

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maple [A]  time = 0.05, size = 112, normalized size = 0.44 \begin {gather*} \frac {2 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (a \,x^{\frac {1}{3}}+b \right ) \left (692835 a^{8} x^{\frac {8}{3}}-583440 a^{7} b \,x^{\frac {7}{3}}+480480 a^{6} b^{2} x^{2}-384384 a^{5} b^{3} x^{\frac {5}{3}}+295680 a^{4} b^{4} x^{\frac {4}{3}}-215040 a^{3} b^{5} x +143360 a^{2} b^{6} x^{\frac {2}{3}}-81920 a \,b^{7} x^{\frac {1}{3}}+32768 b^{8}\right )}{4849845 a^{9} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(a*x+b*x^(2/3))^(3/2)*(a*x^(1/3)+b)*(692835*x^(8/3)*a^8-583440*x^(7/3)*a^7*b+480480*a^6*b^2*x^2-3843
84*x^(5/3)*a^5*b^3+295680*x^(4/3)*a^4*b^4-215040*a^3*b^5*x+143360*x^(2/3)*a^2*b^6-81920*x^(1/3)*a*b^7+32768*b^
8)/x/a^9

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

int(x*(a*x + b*x^(2/3))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \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*(b*x**(2/3)+a*x)**(3/2),x)

[Out]

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

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