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

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

[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)

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Rubi [A]  time = 0.42205, antiderivative size = 255, normalized size of antiderivative = 1., 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} \[ \frac{65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac{32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac{8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac{4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac{512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac{256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac{64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac{32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac{2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \]

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 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])

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])

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*}

Mathematica [A]  time = 0.0855035, size = 135, normalized size = 0.53 \[ \frac{2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt{a x+b x^{2/3}} \left (480480 a^6 b^2 x^2-384384 a^5 b^3 x^{5/3}+295680 a^4 b^4 x^{4/3}+143360 a^2 b^6 x^{2/3}-215040 a^3 b^5 x-583440 a^7 b x^{7/3}+692835 a^8 x^{8/3}-81920 a b^7 \sqrt [3]{x}+32768 b^8\right )}{4849845 a^9 \sqrt [3]{x}} \]

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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Maple [A]  time = 0.004, size = 112, normalized size = 0.4 \begin{align*}{\frac{2}{4849845\,x{a}^{9}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( b+a\sqrt [3]{x} \right ) \left ( 692835\,{x}^{8/3}{a}^{8}-583440\,{x}^{7/3}{a}^{7}b+480480\,{x}^{2}{a}^{6}{b}^{2}-384384\,{x}^{5/3}{a}^{5}{b}^{3}+295680\,{x}^{4/3}{a}^{4}{b}^{4}-215040\,x{a}^{3}{b}^{5}+143360\,{x}^{2/3}{a}^{2}{b}^{6}-81920\,\sqrt [3]{x}a{b}^{7}+32768\,{b}^{8} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(b*x^(2/3)+a*x)^(3/2)*(b+a*x^(1/3))*(692835*x^(8/3)*a^8-583440*x^(7/3)*a^7*b+480480*x^2*a^6*b^2-3843
84*x^(5/3)*a^5*b^3+295680*x^(4/3)*a^4*b^4-215040*x*a^3*b^5+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., size = 0, normalized size = 0. \begin{align*} \int{\left (a x + b x^{\frac{2}{3}}\right )}^{\frac{3}{2}} x\,{d x} \end{align*}

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

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.16902, size = 394, normalized size = 1.55 \begin{align*} -\frac{2}{692835} \, b{\left (\frac{32768 \, b^{\frac{19}{2}}}{a^{9}} - \frac{109395 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} - 978120 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b + 3879876 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{2} - 8953560 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{3} + 13226850 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{4} - 12932920 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{5} + 8314020 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{6} - 3325608 \,{\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}}{a^{9}}\right )} + \frac{2}{1616615} \, a{\left (\frac{65536 \, b^{\frac{21}{2}}}{a^{10}} + \frac{230945 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} - 2297295 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b + 10270260 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{2} - 27159132 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{3} + 47006190 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{4} - 55552770 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{5} + 45265220 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{6} - 24942060 \,{\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} - 1616615 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{9}}{a^{10}}\right )} \end{align*}

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 - (109395*(a*x^(1/3) + b)^(19/2) - 978120*(a*x^(1/3) + b)^(17/2)*b + 3879876*(
a*x^(1/3) + b)^(15/2)*b^2 - 8953560*(a*x^(1/3) + b)^(13/2)*b^3 + 13226850*(a*x^(1/3) + b)^(11/2)*b^4 - 1293292
0*(a*x^(1/3) + b)^(9/2)*b^5 + 8314020*(a*x^(1/3) + b)^(7/2)*b^6 - 3325608*(a*x^(1/3) + b)^(5/2)*b^7 + 692835*(
a*x^(1/3) + b)^(3/2)*b^8)/a^9) + 2/1616615*a*(65536*b^(21/2)/a^10 + (230945*(a*x^(1/3) + b)^(21/2) - 2297295*(
a*x^(1/3) + b)^(19/2)*b + 10270260*(a*x^(1/3) + b)^(17/2)*b^2 - 27159132*(a*x^(1/3) + b)^(15/2)*b^3 + 47006190
*(a*x^(1/3) + b)^(13/2)*b^4 - 55552770*(a*x^(1/3) + b)^(11/2)*b^5 + 45265220*(a*x^(1/3) + b)^(9/2)*b^6 - 24942
060*(a*x^(1/3) + b)^(7/2)*b^7 + 8729721*(a*x^(1/3) + b)^(5/2)*b^8 - 1616615*(a*x^(1/3) + b)^(3/2)*b^9)/a^10)