3.11.41 \(\int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{7/2}} \, dx\) [1041]

Optimal. Leaf size=180 \[ -\frac {139 \sqrt {x} (2+3 x)}{15 \sqrt {2+5 x+3 x^2}}-\frac {4 (3-10 x) \sqrt {2+5 x+3 x^2}}{15 x^{5/2}}+\frac {139 \sqrt {2+5 x+3 x^2}}{15 \sqrt {x}}+\frac {139 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2+5 x+3 x^2}}-\frac {11 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}} \]

[Out]

-139/15*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+139/15*(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)-11*(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)-4/15*(3-10*x)*(3*x^2+5*x+2)^(1
/2)/x^(5/2)+139/15*(3*x^2+5*x+2)^(1/2)/x^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {824, 848, 853, 1203, 1114, 1150} \begin {gather*} -\frac {11 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+\frac {139 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {3 x^2+5 x+2}}+\frac {139 \sqrt {3 x^2+5 x+2}}{15 \sqrt {x}}-\frac {139 \sqrt {x} (3 x+2)}{15 \sqrt {3 x^2+5 x+2}}-\frac {4 \sqrt {3 x^2+5 x+2} (3-10 x)}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 - 5*x)*Sqrt[2 + 5*x + 3*x^2])/x^(7/2),x]

[Out]

(-139*Sqrt[x]*(2 + 3*x))/(15*Sqrt[2 + 5*x + 3*x^2]) - (4*(3 - 10*x)*Sqrt[2 + 5*x + 3*x^2])/(15*x^(5/2)) + (139
*Sqrt[2 + 5*x + 3*x^2])/(15*Sqrt[x]) + (139*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]],
-1/2])/(15*Sqrt[2 + 5*x + 3*x^2]) - (11*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1
/2])/Sqrt[2 + 5*x + 3*x^2]

Rule 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

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 \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{7/2}} \, dx &=-\frac {4 (3-10 x) \sqrt {2+5 x+3 x^2}}{15 x^{5/2}}-\frac {1}{15} \int \frac {139+165 x}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {4 (3-10 x) \sqrt {2+5 x+3 x^2}}{15 x^{5/2}}+\frac {139 \sqrt {2+5 x+3 x^2}}{15 \sqrt {x}}+\frac {1}{15} \int \frac {-165-\frac {417 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {4 (3-10 x) \sqrt {2+5 x+3 x^2}}{15 x^{5/2}}+\frac {139 \sqrt {2+5 x+3 x^2}}{15 \sqrt {x}}+\frac {2}{15} \text {Subst}\left (\int \frac {-165-\frac {417 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 (3-10 x) \sqrt {2+5 x+3 x^2}}{15 x^{5/2}}+\frac {139 \sqrt {2+5 x+3 x^2}}{15 \sqrt {x}}-22 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-\frac {139}{5} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {139 \sqrt {x} (2+3 x)}{15 \sqrt {2+5 x+3 x^2}}-\frac {4 (3-10 x) \sqrt {2+5 x+3 x^2}}{15 x^{5/2}}+\frac {139 \sqrt {2+5 x+3 x^2}}{15 \sqrt {x}}+\frac {139 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2+5 x+3 x^2}}-\frac {11 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\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 = 153, normalized size = 0.85 \begin {gather*} \frac {4 \left (-6+5 x+41 x^2+30 x^3\right )-139 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{7/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-26 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{7/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{15 x^{5/2} \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 - 5*x)*Sqrt[2 + 5*x + 3*x^2])/x^(7/2),x]

[Out]

(4*(-6 + 5*x + 41*x^2 + 30*x^3) - (139*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(7/2)*EllipticE[I*ArcSinh[S
qrt[2/3]/Sqrt[x]], 3/2] - (26*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/
Sqrt[x]], 3/2])/(15*x^(5/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]
time = 0.86, size = 124, normalized size = 0.69

method	result	size
default	\(\frac {87 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}-139 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+2502 x^{4}+4890 x^{3}+2652 x^{2}+120 x -144}{90 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {5}{2}}}\)	\(124\)
risch	\(\frac {417 x^{4}+815 x^{3}+442 x^{2}+20 x -24}{15 x^{\frac {5}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (\frac {11 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {139 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{30 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(198\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {4 \sqrt {3 x^{3}+5 x^{2}+2 x}}{5 x^{3}}+\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{3 x^{2}}+\frac {\frac {139}{5} x^{2}+\frac {139}{3} x +\frac {278}{15}}{\sqrt {x \left (3 x^{2}+5 x +2\right )}}-\frac {11 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {139 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{30 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(227\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-5*x)*(3*x^2+5*x+2)^(1/2)/x^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/90*(87*(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))*x^2-139*(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))*x^2+2502*x^4+4890*x^3+2652*x^2+
120*x-144)/(3*x^2+5*x+2)^(1/2)/x^(5/2)

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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)^(1/2)/x^(7/2),x, algorithm="maxima")

[Out]

-integrate(sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.46, size = 64, normalized size = 0.36 \begin {gather*} -\frac {295 \, \sqrt {3} x^{3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 1251 \, \sqrt {3} x^{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 ) - 9 \, {\left (139 \, x^{2} + 40 \, x - 12\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{135 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-5*x)*(3*x^2+5*x+2)^(1/2)/x^(7/2),x, algorithm="fricas")

[Out]

-1/135*(295*sqrt(3)*x^3*weierstrassPInverse(28/27, 80/729, x + 5/9) - 1251*sqrt(3)*x^3*weierstrassZeta(28/27,
80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 9*(139*x^2 + 40*x - 12)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/
x^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {2 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {7}{2}}}\right )\, dx - \int \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {5}{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)**(1/2)/x**(7/2),x)

[Out]

-Integral(-2*sqrt(3*x**2 + 5*x + 2)/x**(7/2), x) - Integral(5*sqrt(3*x**2 + 5*x + 2)/x**(5/2), 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)^(1/2)/x^(7/2),x, algorithm="giac")

[Out]

integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2}}{x^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(1/2))/x^(7/2),x)

[Out]

int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(1/2))/x^(7/2), x)

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