3.208 \(\int (2 \sqrt {x}-x)^2 x^{3/2} (1+x^2) \, dx\)

Optimal. Leaf size=49 \[ \frac {2 x^{13/2}}{13}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}+\frac {8 x^{7/2}}{7}-\frac {2 x^6}{3}-x^4 \]

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Rubi [A]  time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 1820, 266, 43} \[ \frac {2 x^{13/2}}{13}-\frac {2 x^6}{3}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}-x^4+\frac {8 x^{7/2}}{7} \]

Antiderivative was successfully verified.

[In]

Int[(2*Sqrt[x] - x)^2*x^(3/2)*(1 + x^2),x]

[Out]

(8*x^(7/2))/7 - x^4 + (2*x^(9/2))/9 + (8*x^(11/2))/11 - (2*x^6)/3 + (2*x^(13/2))/13

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {align*} \int \left (2 \sqrt {x}-x\right )^2 x^{3/2} \left (1+x^2\right ) \, dx &=\int \left (2-\sqrt {x}\right )^2 x^{5/2} \left (1+x^2\right ) \, dx\\ &=\int \left (\left (-2+\sqrt {x}\right )^2 x^{5/2}+\left (-2+\sqrt {x}\right )^2 x^{9/2}\right ) \, dx\\ &=\int \left (-2+\sqrt {x}\right )^2 x^{5/2} \, dx+\int \left (-2+\sqrt {x}\right )^2 x^{9/2} \, dx\\ &=2 \operatorname {Subst}\left (\int (-2+x)^2 x^6 \, dx,x,\sqrt {x}\right )+2 \operatorname {Subst}\left (\int (-2+x)^2 x^{10} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (4 x^6-4 x^7+x^8\right ) \, dx,x,\sqrt {x}\right )+2 \operatorname {Subst}\left (\int \left (4 x^{10}-4 x^{11}+x^{12}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {8 x^{7/2}}{7}-x^4+\frac {2 x^{9/2}}{9}+\frac {8 x^{11/2}}{11}-\frac {2 x^6}{3}+\frac {2 x^{13/2}}{13}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.00 \[ \frac {2 x^{13/2}}{13}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}+\frac {8 x^{7/2}}{7}-\frac {2 x^6}{3}-x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[(2*Sqrt[x] - x)^2*x^(3/2)*(1 + x^2),x]

[Out]

(8*x^(7/2))/7 - x^4 + (2*x^(9/2))/9 + (8*x^(11/2))/11 - (2*x^6)/3 + (2*x^(13/2))/13

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IntegrateAlgebraic [A]  time = 0.01, size = 49, normalized size = 1.00 \[ \frac {2 x^{13/2}}{13}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}+\frac {8 x^{7/2}}{7}-\frac {2 x^6}{3}-x^4 \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2*Sqrt[x] - x)^2*x^(3/2)*(1 + x^2),x]

[Out]

(8*x^(7/2))/7 - x^4 + (2*x^(9/2))/9 + (8*x^(11/2))/11 - (2*x^6)/3 + (2*x^(13/2))/13

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fricas [A]  time = 1.63, size = 37, normalized size = 0.76 \[ -\frac {2}{3} \, x^{6} - x^{4} + \frac {2}{9009} \, {\left (693 \, x^{6} + 3276 \, x^{5} + 1001 \, x^{4} + 5148 \, x^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(x^2+1)*(-x+2*x^(1/2))^2,x, algorithm="fricas")

[Out]

-2/3*x^6 - x^4 + 2/9009*(693*x^6 + 3276*x^5 + 1001*x^4 + 5148*x^3)*sqrt(x)

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giac [A]  time = 0.99, size = 31, normalized size = 0.63 \[ \frac {2}{13} \, x^{\frac {13}{2}} - \frac {2}{3} \, x^{6} + \frac {8}{11} \, x^{\frac {11}{2}} + \frac {2}{9} \, x^{\frac {9}{2}} - x^{4} + \frac {8}{7} \, x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*x^(13/2) - 2/3*x^6 + 8/11*x^(11/2) + 2/9*x^(9/2) - x^4 + 8/7*x^(7/2)

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maple [A]  time = 0.29, size = 32, normalized size = 0.65

method	result	size
derivativedivides	\(\frac {8 x^{\frac {7}{2}}}{7}-x^{4}+\frac {2 x^{\frac {9}{2}}}{9}+\frac {8 x^{\frac {11}{2}}}{11}-\frac {2 x^{6}}{3}+\frac {2 x^{\frac {13}{2}}}{13}\)	\(32\)
default	\(\frac {8 x^{\frac {7}{2}}}{7}-x^{4}+\frac {2 x^{\frac {9}{2}}}{9}+\frac {8 x^{\frac {11}{2}}}{11}-\frac {2 x^{6}}{3}+\frac {2 x^{\frac {13}{2}}}{13}\)	\(32\)
trager	\(-\frac {\left (2 x^{5}+2 x^{4}+5 x^{3}+5 x^{2}+5 x +5\right ) \left (-1+x \right )}{3}+\frac {2 x^{\frac {7}{2}} \left (693 x^{3}+3276 x^{2}+1001 x +5148\right )}{9009}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(x^2+1)*(-x+2*x^(1/2))^2,x,method=_RETURNVERBOSE)

[Out]

8/7*x^(7/2)-x^4+2/9*x^(9/2)+8/11*x^(11/2)-2/3*x^6+2/13*x^(13/2)

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maxima [A]  time = 0.53, size = 31, normalized size = 0.63 \[ \frac {2}{13} \, x^{\frac {13}{2}} - \frac {2}{3} \, x^{6} + \frac {8}{11} \, x^{\frac {11}{2}} + \frac {2}{9} \, x^{\frac {9}{2}} - x^{4} + \frac {8}{7} \, x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*x^(13/2) - 2/3*x^6 + 8/11*x^(11/2) + 2/9*x^(9/2) - x^4 + 8/7*x^(7/2)

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mupad [B]  time = 0.19, size = 31, normalized size = 0.63 \[ \frac {8\,x^{7/2}}{7}-\frac {2\,x^6}{3}-x^4+\frac {2\,x^{9/2}}{9}+\frac {8\,x^{11/2}}{11}+\frac {2\,x^{13/2}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*x^(7/2))/7 - (2*x^6)/3 - x^4 + (2*x^(9/2))/9 + (8*x^(11/2))/11 + (2*x^(13/2))/13

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sympy [A]  time = 2.60, size = 42, normalized size = 0.86 \[ \frac {2 x^{\frac {13}{2}}}{13} + \frac {8 x^{\frac {11}{2}}}{11} + \frac {2 x^{\frac {9}{2}}}{9} + \frac {8 x^{\frac {7}{2}}}{7} - \frac {2 x^{6}}{3} - x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(x**2+1)*(-x+2*x**(1/2))**2,x)

[Out]

2*x**(13/2)/13 + 8*x**(11/2)/11 + 2*x**(9/2)/9 + 8*x**(7/2)/7 - 2*x**6/3 - x**4

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