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

Optimal. Leaf size=313 \[ \frac{524288 b^{10} \sqrt{a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac{262144 b^9 \sqrt{a x+b x^{2/3}}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{323323 a^8}+\frac{20480 b^6 x \sqrt{a x+b x^{2/3}}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^2}+\frac{2 x^3 \sqrt{a x+b x^{2/3}}}{7 a} \]

[Out]

(-262144*b^9*Sqrt[b*x^(2/3) + a*x])/(323323*a^10) + (524288*b^10*Sqrt[b*x^(2/3) + a*x])/(323323*a^11*x^(1/3))
+ (196608*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(323323*a^9) - (163840*b^7*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(323323
*a^8) + (20480*b^6*x*Sqrt[b*x^(2/3) + a*x])/(46189*a^7) - (18432*b^5*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(46189*a^6
) + (1536*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^5) - (768*b^3*x^2*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) + (72
0*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^3) - (40*b*x^(8/3)*Sqrt[b*x^(2/3) + a*x])/(133*a^2) + (2*x^3*Sqrt
[b*x^(2/3) + a*x])/(7*a)

________________________________________________________________________________________

Rubi [A]  time = 0.53092, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ \frac{524288 b^{10} \sqrt{a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac{262144 b^9 \sqrt{a x+b x^{2/3}}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{323323 a^8}+\frac{20480 b^6 x \sqrt{a x+b x^{2/3}}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^2}+\frac{2 x^3 \sqrt{a x+b x^{2/3}}}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(-262144*b^9*Sqrt[b*x^(2/3) + a*x])/(323323*a^10) + (524288*b^10*Sqrt[b*x^(2/3) + a*x])/(323323*a^11*x^(1/3))
+ (196608*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(323323*a^9) - (163840*b^7*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(323323
*a^8) + (20480*b^6*x*Sqrt[b*x^(2/3) + a*x])/(46189*a^7) - (18432*b^5*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(46189*a^6
) + (1536*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^5) - (768*b^3*x^2*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) + (72
0*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^3) - (40*b*x^(8/3)*Sqrt[b*x^(2/3) + a*x])/(133*a^2) + (2*x^3*Sqrt
[b*x^(2/3) + a*x])/(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 \frac{x^3}{\sqrt{b x^{2/3}+a x}} \, dx &=\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{(20 b) \int \frac{x^{8/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{21 a}\\ &=-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (120 b^2\right ) \int \frac{x^{7/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{133 a^2}\\ &=\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (1920 b^3\right ) \int \frac{x^2}{\sqrt{b x^{2/3}+a x}} \, dx}{2261 a^3}\\ &=-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (256 b^4\right ) \int \frac{x^{5/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{323 a^4}\\ &=\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (3072 b^5\right ) \int \frac{x^{4/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{4199 a^5}\\ &=-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (30720 b^6\right ) \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx}{46189 a^6}\\ &=\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (81920 b^7\right ) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{138567 a^7}\\ &=-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (163840 b^8\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{323323 a^8}\\ &=\frac{196608 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}-\frac{\left (131072 b^9\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{323323 a^9}\\ &=-\frac{262144 b^9 \sqrt{b x^{2/3}+a x}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}+\frac{\left (262144 b^{10}\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{969969 a^{10}}\\ &=-\frac{262144 b^9 \sqrt{b x^{2/3}+a x}}{323323 a^{10}}+\frac{524288 b^{10} \sqrt{b x^{2/3}+a x}}{323323 a^{11} \sqrt [3]{x}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{b x^{2/3}+a x}}{323323 a^8}+\frac{20480 b^6 x \sqrt{b x^{2/3}+a x}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{b x^{2/3}+a x}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{b x^{2/3}+a x}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{b x^{2/3}+a x}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{b x^{2/3}+a x}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{b x^{2/3}+a x}}{133 a^2}+\frac{2 x^3 \sqrt{b x^{2/3}+a x}}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.115059, size = 148, normalized size = 0.47 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (51480 a^8 b^2 x^{8/3}-54912 a^7 b^3 x^{7/3}+59136 a^6 b^4 x^2-64512 a^5 b^5 x^{5/3}+71680 a^4 b^6 x^{4/3}+98304 a^2 b^8 x^{2/3}-81920 a^3 b^7 x-48620 a^9 b x^3+46189 a^{10} x^{10/3}-131072 a b^9 \sqrt [3]{x}+262144 b^{10}\right )}{323323 a^{11} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(262144*b^10 - 131072*a*b^9*x^(1/3) + 98304*a^2*b^8*x^(2/3) - 81920*a^3*b^7*x + 71680
*a^4*b^6*x^(4/3) - 64512*a^5*b^5*x^(5/3) + 59136*a^6*b^4*x^2 - 54912*a^7*b^3*x^(7/3) + 51480*a^8*b^2*x^(8/3) -
48620*a^9*b*x^3 + 46189*a^10*x^(10/3)))/(323323*a^11*x^(1/3))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 134, normalized size = 0.4 \begin{align*}{\frac{2}{323323\,{a}^{11}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 46189\,{x}^{10/3}{a}^{10}-48620\,{x}^{3}{a}^{9}b+51480\,{x}^{8/3}{a}^{8}{b}^{2}-54912\,{x}^{7/3}{a}^{7}{b}^{3}+59136\,{x}^{2}{a}^{6}{b}^{4}-64512\,{x}^{5/3}{a}^{5}{b}^{5}+71680\,{x}^{4/3}{a}^{4}{b}^{6}-81920\,x{a}^{3}{b}^{7}+98304\,{x}^{2/3}{a}^{2}{b}^{8}-131072\,\sqrt [3]{x}a{b}^{9}+262144\,{b}^{10} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/323323*x^(1/3)*(b+a*x^(1/3))*(46189*x^(10/3)*a^10-48620*x^3*a^9*b+51480*x^(8/3)*a^8*b^2-54912*x^(7/3)*a^7*b^
3+59136*x^2*a^6*b^4-64512*x^(5/3)*a^5*b^5+71680*x^(4/3)*a^4*b^6-81920*x*a^3*b^7+98304*x^(2/3)*a^2*b^8-131072*x
^(1/3)*a*b^9+262144*b^10)/(b*x^(2/3)+a*x)^(1/2)/a^11

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a x + b x^{\frac{2}{3}}}}\,{d x} \end{align*}

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(x^3/sqrt(a*x + b*x^(2/3)), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{a x + b x^{\frac{2}{3}}}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.15723, size = 221, normalized size = 0.71 \begin{align*} -\frac{524288 \, b^{\frac{21}{2}}}{323323 \, a^{11}} + \frac{2 \,{\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 )}}{323323 \, a^{11}} \end{align*}

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]

-524288/323323*b^(21/2)/a^11 + 2/323323*(46189*(a*x^(1/3) + b)^(21/2) - 510510*(a*x^(1/3) + b)^(19/2)*b + 2567
565*(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 - 22
221108*(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^11