3.2.96 \(\int \frac {x^2}{(b x^{2/3}+a x)^{3/2}} \, dx\) [196]

Optimal. Leaf size=160 \[ -\frac {6 x^2}{a \sqrt {b x^{2/3}+a x}}-\frac {256 b^3 \sqrt {b x^{2/3}+a x}}{21 a^5}+\frac {512 b^4 \sqrt {b x^{2/3}+a x}}{21 a^6 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{7 a^4}-\frac {160 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^3}+\frac {20 x \sqrt {b x^{2/3}+a x}}{3 a^2} \]

[Out]

-6*x^2/a/(b*x^(2/3)+a*x)^(1/2)-256/21*b^3*(b*x^(2/3)+a*x)^(1/2)/a^5+512/21*b^4*(b*x^(2/3)+a*x)^(1/2)/a^6/x^(1/
3)+64/7*b^2*x^(1/3)*(b*x^(2/3)+a*x)^(1/2)/a^4-160/21*b*x^(2/3)*(b*x^(2/3)+a*x)^(1/2)/a^3+20/3*x*(b*x^(2/3)+a*x
)^(1/2)/a^2

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Rubi [A]
time = 0.16, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2040, 2041, 2027, 2039} \begin {gather*} \frac {512 b^4 \sqrt {a x+b x^{2/3}}}{21 a^6 \sqrt [3]{x}}-\frac {256 b^3 \sqrt {a x+b x^{2/3}}}{21 a^5}+\frac {64 b^2 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{7 a^4}-\frac {160 b x^{2/3} \sqrt {a x+b x^{2/3}}}{21 a^3}+\frac {20 x \sqrt {a x+b x^{2/3}}}{3 a^2}-\frac {6 x^2}{a \sqrt {a x+b x^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-6*x^2)/(a*Sqrt[b*x^(2/3) + a*x]) - (256*b^3*Sqrt[b*x^(2/3) + a*x])/(21*a^5) + (512*b^4*Sqrt[b*x^(2/3) + a*x]
)/(21*a^6*x^(1/3)) + (64*b^2*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(7*a^4) - (160*b*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(2
1*a^3) + (20*x*Sqrt[b*x^(2/3) + a*x])/(3*a^2)

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 2040

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] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))
, Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n,
j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (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 \frac {x^2}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=-\frac {6 x^2}{a \sqrt {b x^{2/3}+a x}}+\frac {10 \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{a}\\ &=-\frac {6 x^2}{a \sqrt {b x^{2/3}+a x}}+\frac {20 x \sqrt {b x^{2/3}+a x}}{3 a^2}-\frac {(80 b) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{9 a^2}\\ &=-\frac {6 x^2}{a \sqrt {b x^{2/3}+a x}}-\frac {160 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^3}+\frac {20 x \sqrt {b x^{2/3}+a x}}{3 a^2}+\frac {\left (160 b^2\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{21 a^3}\\ &=-\frac {6 x^2}{a \sqrt {b x^{2/3}+a x}}+\frac {64 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{7 a^4}-\frac {160 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^3}+\frac {20 x \sqrt {b x^{2/3}+a x}}{3 a^2}-\frac {\left (128 b^3\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{21 a^4}\\ &=-\frac {6 x^2}{a \sqrt {b x^{2/3}+a x}}-\frac {256 b^3 \sqrt {b x^{2/3}+a x}}{21 a^5}+\frac {64 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{7 a^4}-\frac {160 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^3}+\frac {20 x \sqrt {b x^{2/3}+a x}}{3 a^2}+\frac {\left (256 b^4\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{63 a^5}\\ &=-\frac {6 x^2}{a \sqrt {b x^{2/3}+a x}}-\frac {256 b^3 \sqrt {b x^{2/3}+a x}}{21 a^5}+\frac {512 b^4 \sqrt {b x^{2/3}+a x}}{21 a^6 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{7 a^4}-\frac {160 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^3}+\frac {20 x \sqrt {b x^{2/3}+a x}}{3 a^2}\\ \end {align*}

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Mathematica [A]
time = 4.37, size = 85, normalized size = 0.53 \begin {gather*} \frac {512 b^5 \sqrt [3]{x}+256 a b^4 x^{2/3}-64 a^2 b^3 x+32 a^3 b^2 x^{4/3}-20 a^4 b x^{5/3}+14 a^5 x^2}{21 a^6 \sqrt {b x^{2/3}+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(512*b^5*x^(1/3) + 256*a*b^4*x^(2/3) - 64*a^2*b^3*x + 32*a^3*b^2*x^(4/3) - 20*a^4*b*x^(5/3) + 14*a^5*x^2)/(21*
a^6*Sqrt[b*x^(2/3) + a*x])

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Maple [A]
time = 0.36, size = 77, normalized size = 0.48

method	result	size
derivativedivides	\(\frac {2 x \left (b +a \,x^{\frac {1}{3}}\right ) \left (7 a^{5} x^{\frac {5}{3}}-10 a^{4} b \,x^{\frac {4}{3}}+16 a^{3} b^{2} x -32 a^{2} b^{3} x^{\frac {2}{3}}+128 a \,b^{4} x^{\frac {1}{3}}+256 b^{5}\right )}{21 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{6}}\)	\(77\)
default	\(\frac {2 x \left (b +a \,x^{\frac {1}{3}}\right ) \left (7 a^{5} x^{\frac {5}{3}}-10 a^{4} b \,x^{\frac {4}{3}}+16 a^{3} b^{2} x -32 a^{2} b^{3} x^{\frac {2}{3}}+128 a \,b^{4} x^{\frac {1}{3}}+256 b^{5}\right )}{21 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{6}}\)	\(77\)

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]

2/21*x*(b+a*x^(1/3))*(7*a^5*x^(5/3)-10*a^4*b*x^(4/3)+16*a^3*b^2*x-32*a^2*b^3*x^(2/3)+128*a*b^4*x^(1/3)+256*b^5
)/(b*x^(2/3)+a*x)^(3/2)/a^6

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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(x^2/(a*x + b*x^(2/3))^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1598 vs. \(2 (120) = 240\).
time = 192.10, size = 1598, normalized size = 9.99 \begin {gather*} \text {Too large to display} \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]

-1/21*((3145728*a^3*b^13 + 2621440*a^3*b^12 - 983040*a^3*b^11 - 10192*a^12 + 196608*(17*a^6 - 3*a^3)*b^10 + 40
96*(464*a^6 + 53*a^3)*b^9 - 6144*(246*a^6 + a^3)*b^8 + 768*(1120*a^9 - 2560*a^6 - 3*a^3)*b^7 - 256*(548*a^9 -
1569*a^6)*b^6 - 768*(1477*a^9 + 31*a^6)*b^5 - 48*(2304*a^12 + 21176*a^9 + 33*a^6)*b^4 - 4032*(96*a^12 - 23*a^9
)*b^3 - 12*(27648*a^12 + 527*a^9)*b^2 + 3*(39296*a^12 + 51*a^9)*b)*x^2 + (3145728*b^16 + 2621440*b^15 + 196608
*(17*a^3 - 3)*b^13 - 983040*b^14 + 4096*(464*a^3 + 53)*b^12 - 10192*a^9*b^3 - 6144*(246*a^3 + 1)*b^11 + 768*(1
120*a^6 - 2560*a^3 - 3)*b^10 - 256*(548*a^6 - 1569*a^3)*b^9 - 768*(1477*a^6 + 31*a^3)*b^8 - 48*(2304*a^9 + 211
76*a^6 + 33*a^3)*b^7 - 4032*(96*a^9 - 23*a^6)*b^6 - 12*(27648*a^9 + 527*a^6)*b^5 + 3*(39296*a^9 + 51*a^6)*b^4)
*x - 2*(7*(4096*a^7*b^9 + 6144*a^7*b^8 + 768*a^7*b^7 - 4096*a^13 - 144*a^10*b^2 + 216*a^10*b - 27*a^10 + 256*(
16*a^10 - 7*a^7)*b^6 + 48*(128*a^10 - 3*a^7)*b^5 + 24*(32*a^10 + 9*a^7)*b^4 - (5888*a^10 + 27*a^7)*b^3)*x^3 -
58*(4096*a^4*b^12 + 6144*a^4*b^11 + 768*a^4*b^10 - 144*a^7*b^5 + 216*a^7*b^4 + 256*(16*a^7 - 7*a^4)*b^9 + 48*(
128*a^7 - 3*a^4)*b^8 + 24*(32*a^7 + 9*a^4)*b^7 - (5888*a^7 + 27*a^4)*b^6 - (4096*a^10 + 27*a^7)*b^3)*x^2 - 128
*(4096*a*b^15 + 6144*a*b^14 + 768*a*b^13 + 256*(16*a^4 - 7*a)*b^12 - 144*a^4*b^8 + 48*(128*a^4 - 3*a)*b^11 + 2
16*a^4*b^7 + 24*(32*a^4 + 9*a)*b^10 - (5888*a^4 + 27*a)*b^9 - (4096*a^7 + 27*a^4)*b^6)*x + (1048576*b^16 + 157
2864*b^15 + 65536*(16*a^3 - 7)*b^13 + 196608*b^14 + 12288*(128*a^3 - 3)*b^12 - 36864*a^3*b^9 + 6144*(32*a^3 +
9)*b^11 + 55296*a^3*b^8 - 256*(5888*a^3 + 27)*b^10 - 256*(4096*a^6 + 27*a^3)*b^7 - 17*(4096*a^6*b^10 + 6144*a^
6*b^9 + 768*a^6*b^8 - 144*a^9*b^3 + 216*a^9*b^2 + 256*(16*a^9 - 7*a^6)*b^7 + 48*(128*a^9 - 3*a^6)*b^6 + 24*(32
*a^9 + 9*a^6)*b^5 - (5888*a^9 + 27*a^6)*b^4 - (4096*a^12 + 27*a^9)*b)*x^2 + 176*(4096*a^3*b^13 + 6144*a^3*b^12
+ 768*a^3*b^11 - 144*a^6*b^6 + 216*a^6*b^5 + 256*(16*a^6 - 7*a^3)*b^10 + 48*(128*a^6 - 3*a^3)*b^9 + 24*(32*a^
6 + 9*a^3)*b^8 - (5888*a^6 + 27*a^3)*b^7 - (4096*a^9 + 27*a^6)*b^4)*x)*x^(2/3) + 3*(11*(4096*a^5*b^11 + 6144*a
^5*b^10 + 768*a^5*b^9 - 144*a^8*b^4 + 216*a^8*b^3 + 256*(16*a^8 - 7*a^5)*b^8 + 48*(128*a^8 - 3*a^5)*b^7 + 24*(
32*a^8 + 9*a^5)*b^6 - (5888*a^8 + 27*a^5)*b^5 - (4096*a^11 + 27*a^8)*b^2)*x^2 + 32*(4096*a^2*b^14 + 6144*a^2*b
^13 + 768*a^2*b^12 - 144*a^5*b^7 + 256*(16*a^5 - 7*a^2)*b^11 + 216*a^5*b^6 + 48*(128*a^5 - 3*a^2)*b^10 + 24*(3
2*a^5 + 9*a^2)*b^9 - (5888*a^5 + 27*a^2)*b^8 - (4096*a^8 + 27*a^5)*b^5)*x)*x^(1/3))*sqrt(a*x + b*x^(2/3)))/((4
096*a^9*b^9 + 6144*a^9*b^8 + 768*a^9*b^7 - 4096*a^15 - 144*a^12*b^2 + 216*a^12*b - 27*a^12 + 256*(16*a^12 - 7*
a^9)*b^6 + 48*(128*a^12 - 3*a^9)*b^5 + 24*(32*a^12 + 9*a^9)*b^4 - (5888*a^12 + 27*a^9)*b^3)*x^2 + (4096*a^6*b^
12 + 6144*a^6*b^11 + 768*a^6*b^10 - 144*a^9*b^5 + 216*a^9*b^4 + 256*(16*a^9 - 7*a^6)*b^9 + 48*(128*a^9 - 3*a^6
)*b^8 + 24*(32*a^9 + 9*a^6)*b^7 - (5888*a^9 + 27*a^6)*b^6 - (4096*a^12 + 27*a^9)*b^3)*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {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)

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Giac [A]
time = 1.94, size = 112, normalized size = 0.70 \begin {gather*} -\frac {512 \, b^{\frac {9}{2}}}{21 \, a^{6}} + \frac {6 \, b^{5}}{\sqrt {a x^{\frac {1}{3}} + b} a^{6}} + \frac {2 \, {\left (7 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{48} - 45 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{48} b + 126 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{48} b^{2} - 210 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{48} b^{3} + 315 \, \sqrt {a x^{\frac {1}{3}} + b} a^{48} b^{4}\right )}}{21 \, a^{54}} \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]

-512/21*b^(9/2)/a^6 + 6*b^5/(sqrt(a*x^(1/3) + b)*a^6) + 2/21*(7*(a*x^(1/3) + b)^(9/2)*a^48 - 45*(a*x^(1/3) + b
)^(7/2)*a^48*b + 126*(a*x^(1/3) + b)^(5/2)*a^48*b^2 - 210*(a*x^(1/3) + b)^(3/2)*a^48*b^3 + 315*sqrt(a*x^(1/3)
+ b)*a^48*b^4)/a^54

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {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)

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