3.11.46 \(\int (2-5 x) \sqrt {x} (2+5 x+3 x^2)^{3/2} \, dx\) [1046]

Optimal. Leaf size=210 \[ -\frac {424 \sqrt {x} (2+3 x)}{1155 \sqrt {2+5 x+3 x^2}}-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {424 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1155 \sqrt {2+5 x+3 x^2}}-\frac {36 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{77 \sqrt {2+5 x+3 x^2}} \]

[Out]

4/231*(65+84*x)*(3*x^2+5*x+2)^(3/2)*x^(1/2)-10/33*(3*x^2+5*x+2)^(5/2)*x^(1/2)-424/1155*(2+3*x)*x^(1/2)/(3*x^2+
5*x+2)^(1/2)+424/1155*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x
)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-36/77*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticF(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/
2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-4/385*(55+39*x)*x^(1/2)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {846, 828, 853, 1203, 1114, 1150} \begin {gather*} -\frac {36 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{77 \sqrt {3 x^2+5 x+2}}+\frac {424 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1155 \sqrt {3 x^2+5 x+2}}-\frac {10}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}+\frac {4}{231} \sqrt {x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac {4}{385} \sqrt {x} (39 x+55) \sqrt {3 x^2+5 x+2}-\frac {424 \sqrt {x} (3 x+2)}{1155 \sqrt {3 x^2+5 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 5*x)*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2),x]

[Out]

(-424*Sqrt[x]*(2 + 3*x))/(1155*Sqrt[2 + 5*x + 3*x^2]) - (4*Sqrt[x]*(55 + 39*x)*Sqrt[2 + 5*x + 3*x^2])/385 + (4
*Sqrt[x]*(65 + 84*x)*(2 + 5*x + 3*x^2)^(3/2))/231 - (10*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/33 + (424*Sqrt[2]*(1
+ x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(1155*Sqrt[2 + 5*x + 3*x^2]) - (36*Sqrt[2]*(1 +
x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(77*Sqrt[2 + 5*x + 3*x^2])

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 853

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2, Subst[Int[(f +
g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1114

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(2*a + (b - q
)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*Elli
pticF[ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^
2 - 4*a*c, 0]

Rule 1150

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b -
q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (
b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(
q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1203

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e, Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b +
q)/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int (2-5 x) \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{33} \int \frac {(5+108 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {4 \int \frac {\left (270+\frac {1053 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx}{2079}\\ &=-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {8 \int \frac {-\frac {10935}{2}-\frac {12879 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{93555}\\ &=-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {16 \text {Subst}\left (\int \frac {-\frac {10935}{2}-\frac {12879 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{93555}\\ &=-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {72}{77} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-\frac {424}{385} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {424 \sqrt {x} (2+3 x)}{1155 \sqrt {2+5 x+3 x^2}}-\frac {4}{385} \sqrt {x} (55+39 x) \sqrt {2+5 x+3 x^2}+\frac {4}{231} \sqrt {x} (65+84 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {10}{33} \sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {424 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{1155 \sqrt {2+5 x+3 x^2}}-\frac {36 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{77 \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.15, size = 173, normalized size = 0.82 \begin {gather*} \frac {-424 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (424+520 x-3106 x^2-6140 x^3+3497 x^4+17775 x^5+16065 x^6+4725 x^7+58 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )\right )}{1155 \sqrt {x} \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 5*x)*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2),x]

[Out]

((-424*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - 2*(424
+ 520*x - 3106*x^2 - 6140*x^3 + 3497*x^4 + 17775*x^5 + 16065*x^6 + 4725*x^7 + (58*I)*Sqrt[2]*Sqrt[1 + x^(-1)]
*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2]))/(1155*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]
time = 0.73, size = 132, normalized size = 0.63

method	result	size
default	\(\frac {-\frac {90 x^{7}}{11}-\frac {306 x^{6}}{11}-\frac {2370 x^{5}}{77}+\frac {32 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{1155}-\frac {212 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3465}-\frac {6994 x^{4}}{1155}+\frac {2456 x^{3}}{231}+\frac {7484 x^{2}}{1155}+\frac {72 x}{77}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(132\)
risch	\(-\frac {2 \left (1575 x^{4}+2730 x^{3}+325 x^{2}-1196 x -270\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{1155}-\frac {\left (\frac {12 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{77 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {212 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{1155 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(198\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {30 x^{4} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11}-\frac {52 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11}-\frac {130 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{231}+\frac {2392 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{1155}+\frac {36 \sqrt {3 x^{3}+5 x^{2}+2 x}}{77}-\frac {12 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{77 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {212 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{1155 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(259\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-5*x)*(3*x^2+5*x+2)^(3/2)*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3465/(3*x^2+5*x+2)^(1/2)/x^(1/2)*(-14175*x^7-48195*x^6-53325*x^5+48*(6*x+4)^(1/2)*(3*x+3)^(1/2)*6^(1/2)*(-x)
^(1/2)*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-106*(6*x+4)^(1/2)*(3*x+3)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticE(1/2
*(6*x+4)^(1/2),I*2^(1/2))-10491*x^4+18420*x^3+11226*x^2+1620*x)

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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((2-5*x)*(3*x^2+5*x+2)^(3/2)*x^(1/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)*sqrt(x), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.71, size = 63, normalized size = 0.30 \begin {gather*} -\frac {2}{1155} \, {\left (1575 \, x^{4} + 2730 \, x^{3} + 325 \, x^{2} - 1196 \, x - 270\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {32}{297} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {424}{1155} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)*x^(1/2),x, algorithm="fricas")

[Out]

-2/1155*(1575*x^4 + 2730*x^3 + 325*x^2 - 1196*x - 270)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) - 32/297*sqrt(3)*weierstr
assPInverse(28/27, 80/729, x + 5/9) + 424/1155*sqrt(3)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/2
7, 80/729, x + 5/9))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 4 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 19 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-5*x)*(3*x**2+5*x+2)**(3/2)*x**(1/2),x)

[Out]

-Integral(-4*sqrt(x)*sqrt(3*x**2 + 5*x + 2), x) - Integral(19*x**(5/2)*sqrt(3*x**2 + 5*x + 2), x) - Integral(1
5*x**(7/2)*sqrt(3*x**2 + 5*x + 2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)*x^(1/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)*sqrt(x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \sqrt {x}\,\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^(1/2)*(5*x - 2)*(5*x + 3*x^2 + 2)^(3/2),x)

[Out]

-int(x^(1/2)*(5*x - 2)*(5*x + 3*x^2 + 2)^(3/2), x)

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