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

Optimal. Leaf size=225 \[ -\frac {4096 b^7 \sqrt {a x+b x^{2/3}}}{2145 a^8 \sqrt [3]{x}}+\frac {2048 b^6 \sqrt {a x+b x^{2/3}}}{2145 a^7}-\frac {512 b^5 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {a x+b x^{2/3}}}{429 a^5}-\frac {224 b^3 x \sqrt {a x+b x^{2/3}}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {a x+b x^{2/3}}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {a x+b x^{2/3}}}{65 a^2}+\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a} \]

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Rubi [A]  time = 0.35, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \begin {gather*} -\frac {4096 b^7 \sqrt {a x+b x^{2/3}}}{2145 a^8 \sqrt [3]{x}}+\frac {2048 b^6 \sqrt {a x+b x^{2/3}}}{2145 a^7}-\frac {512 b^5 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {a x+b x^{2/3}}}{429 a^5}-\frac {224 b^3 x \sqrt {a x+b x^{2/3}}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {a x+b x^{2/3}}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {a x+b x^{2/3}}}{65 a^2}+\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(2048*b^6*Sqrt[b*x^(2/3) + a*x])/(2145*a^7) - (4096*b^7*Sqrt[b*x^(2/3) + a*x])/(2145*a^8*x^(1/3)) - (512*b^5*x
^(1/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^6) + (256*b^4*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(429*a^5) - (224*b^3*x*Sqrt[
b*x^(2/3) + a*x])/(429*a^4) + (336*b^2*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^3) - (28*b*x^(5/3)*Sqrt[b*x^(2/3)
+ a*x])/(65*a^2) + (2*x^2*Sqrt[b*x^(2/3) + a*x])/(5*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 \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx &=\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {(14 b) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{15 a}\\ &=-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}+\frac {\left (56 b^2\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{65 a^2}\\ &=\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {\left (112 b^3\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{143 a^3}\\ &=-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}+\frac {\left (896 b^4\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{1287 a^4}\\ &=\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {\left (256 b^5\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{429 a^5}\\ &=-\frac {512 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}+\frac {\left (1024 b^6\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{2145 a^6}\\ &=\frac {2048 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^7}-\frac {512 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}-\frac {\left (2048 b^7\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{6435 a^7}\\ &=\frac {2048 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^7}-\frac {4096 b^7 \sqrt {b x^{2/3}+a x}}{2145 a^8 \sqrt [3]{x}}-\frac {512 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 111, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {a x+b x^{2/3}} \left (429 a^7 x^{7/3}-462 a^6 b x^2+504 a^5 b^2 x^{5/3}-560 a^4 b^3 x^{4/3}+640 a^3 b^4 x-768 a^2 b^5 x^{2/3}+1024 a b^6 \sqrt [3]{x}-2048 b^7\right )}{2145 a^8 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(-2048*b^7 + 1024*a*b^6*x^(1/3) - 768*a^2*b^5*x^(2/3) + 640*a^3*b^4*x - 560*a^4*b^3*x
^(4/3) + 504*a^5*b^2*x^(5/3) - 462*a^6*b*x^2 + 429*a^7*x^(7/3)))/(2145*a^8*x^(1/3))

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IntegrateAlgebraic [A]  time = 0.10, size = 111, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {a x+b x^{2/3}} \left (429 a^7 x^{7/3}-462 a^6 b x^2+504 a^5 b^2 x^{5/3}-560 a^4 b^3 x^{4/3}+640 a^3 b^4 x-768 a^2 b^5 x^{2/3}+1024 a b^6 \sqrt [3]{x}-2048 b^7\right )}{2145 a^8 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(-2048*b^7 + 1024*a*b^6*x^(1/3) - 768*a^2*b^5*x^(2/3) + 640*a^3*b^4*x - 560*a^4*b^3*x
^(4/3) + 504*a^5*b^2*x^(5/3) - 462*a^6*b*x^2 + 429*a^7*x^(7/3)))/(2145*a^8*x^(1/3))

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

[Out]

Timed out

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giac [A]  time = 0.27, size = 122, normalized size = 0.54 \begin {gather*} \frac {4096 \, b^{\frac {15}{2}}}{2145 \, a^{8}} + \frac {2 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} - 3465 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b + 12285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{2} - 25025 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{3} + 32175 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{4} - 27027 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{6} - 6435 \, \sqrt {a x^{\frac {1}{3}} + b} b^{7}\right )}}{2145 \, a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4096/2145*b^(15/2)/a^8 + 2/2145*(429*(a*x^(1/3) + b)^(15/2) - 3465*(a*x^(1/3) + b)^(13/2)*b + 12285*(a*x^(1/3)
+ b)^(11/2)*b^2 - 25025*(a*x^(1/3) + b)^(9/2)*b^3 + 32175*(a*x^(1/3) + b)^(7/2)*b^4 - 27027*(a*x^(1/3) + b)^(
5/2)*b^5 + 15015*(a*x^(1/3) + b)^(3/2)*b^6 - 6435*sqrt(a*x^(1/3) + b)*b^7)/a^8

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maple [A]  time = 0.05, size = 101, normalized size = 0.45 \begin {gather*} \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (429 a^{7} x^{\frac {7}{3}}-462 a^{6} b \,x^{2}+504 a^{5} b^{2} x^{\frac {5}{3}}-560 a^{4} b^{3} x^{\frac {4}{3}}+640 a^{3} b^{4} x -768 a^{2} b^{5} x^{\frac {2}{3}}+1024 a \,b^{6} x^{\frac {1}{3}}-2048 b^{7}\right ) x^{\frac {1}{3}}}{2145 \sqrt {a x +b \,x^{\frac {2}{3}}}\, a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2145*x^(1/3)*(a*x^(1/3)+b)*(429*x^(7/3)*a^7-462*x^2*a^6*b+504*x^(5/3)*a^5*b^2-560*a^4*x^(4/3)*b^3+640*x*a^3*
b^4-768*x^(2/3)*a^2*b^5+1024*x^(1/3)*a*b^6-2048*b^7)/(a*x+b*x^(2/3))^(1/2)/a^8

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x + b*x^(2/3))^(1/2),x)

[Out]

int(x^2/(a*x + b*x^(2/3))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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