3.26.93 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^{13/2}} \, dx\) [2593]

Optimal. Leaf size=229 \[ \frac {14807 \sqrt {2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac {5861 \sqrt {2+5 x+3 x^2}}{618750 \sqrt {3+2 x}}-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac {5861 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{412500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {14807 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{577500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

1/495*(258+367*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2)-5861/1237500*EllipticE(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))
*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+14807/1732500*EllipticF(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(
-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+14807/866250*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)-1/57750*(15647+
14773*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(7/2)+5861/618750*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {824, 848, 857, 732, 435, 430} \begin {gather*} \frac {14807 \sqrt {-3 x^2-5 x-2} F\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{577500 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {5861 \sqrt {-3 x^2-5 x-2} E\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{412500 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}-\frac {(14773 x+15647) \sqrt {3 x^2+5 x+2}}{57750 (2 x+3)^{7/2}}+\frac {5861 \sqrt {3 x^2+5 x+2}}{618750 \sqrt {2 x+3}}+\frac {14807 \sqrt {3 x^2+5 x+2}}{866250 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(13/2),x]

[Out]

(14807*Sqrt[2 + 5*x + 3*x^2])/(866250*(3 + 2*x)^(3/2)) + (5861*Sqrt[2 + 5*x + 3*x^2])/(618750*Sqrt[3 + 2*x]) -
((15647 + 14773*x)*Sqrt[2 + 5*x + 3*x^2])/(57750*(3 + 2*x)^(7/2)) + ((258 + 367*x)*(2 + 5*x + 3*x^2)^(3/2))/(
495*(3 + 2*x)^(11/2)) - (5861*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(412500*Sqr
t[3]*Sqrt[2 + 5*x + 3*x^2]) + (14807*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(577
500*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

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 857

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

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx &=\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac {1}{330} \int \frac {(-194-303 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx\\ &=-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}+\frac {\int \frac {12185+13059 x}{(3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \, dx}{115500}\\ &=\frac {14807 \sqrt {2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac {\int \frac {-23059-\frac {44421 x}{2}}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{866250}\\ &=\frac {14807 \sqrt {2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac {5861 \sqrt {2+5 x+3 x^2}}{618750 \sqrt {3+2 x}}-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}+\frac {\int \frac {-\frac {73569}{4}-\frac {123081 x}{4}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{2165625}\\ &=\frac {14807 \sqrt {2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac {5861 \sqrt {2+5 x+3 x^2}}{618750 \sqrt {3+2 x}}-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac {5861 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{825000}+\frac {14807 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{1155000}\\ &=\frac {14807 \sqrt {2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac {5861 \sqrt {2+5 x+3 x^2}}{618750 \sqrt {3+2 x}}-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac {\left (5861 \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{412500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (14807 \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{577500 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {14807 \sqrt {2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac {5861 \sqrt {2+5 x+3 x^2}}{618750 \sqrt {3+2 x}}-\frac {(15647+14773 x) \sqrt {2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac {(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac {5861 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{412500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {14807 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{577500 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]
time = 30.35, size = 227, normalized size = 0.99 \begin {gather*} -\frac {-4 \left (2+5 x+3 x^2\right ) \left (9919671+42879355 x+65139670 x^2+41848650 x^3+11031040 x^4+1312864 x^5\right )+2 (3+2 x)^5 \left (82054 \left (2+5 x+3 x^2\right )+41027 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+3394 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )\right )}{17325000 (3+2 x)^{11/2} \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(13/2),x]

[Out]

-1/17325000*(-4*(2 + 5*x + 3*x^2)*(9919671 + 42879355*x + 65139670*x^2 + 41848650*x^3 + 11031040*x^4 + 1312864
*x^5) + 2*(3 + 2*x)^5*(82054*(2 + 5*x + 3*x^2) + 41027*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2
+ 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 3394*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 +
2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/((3 + 2*x)^(11/2)*Sqrt
[2 + 5*x + 3*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(574\) vs. \(2(185)=370\).
time = 0.06, size = 575, normalized size = 2.51

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5632 \left (x +\frac {3}{2}\right )^{6}}+\frac {1303 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{25344 \left (x +\frac {3}{2}\right )^{5}}-\frac {2701 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{40320 \left (x +\frac {3}{2}\right )^{4}}+\frac {34679 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1848000 \left (x +\frac {3}{2}\right )^{3}}+\frac {14807 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{3465000 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {5861}{206250} x^{2}+\frac {5861}{123750} x +\frac {5861}{309375}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {24523 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{43312500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {5861 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{6187500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(318\)
default	\(\frac {1312864 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+1056256 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+9846480 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{4} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+7921920 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{4} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{3} \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \sqrt {3+2 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{3} \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \sqrt {3+2 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] x^{7}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}[phone] x^{6}+9969561 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+8020944 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{86625000 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {11}{2}}}\)	\(575\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x,method=_RETURNVERBOSE)

[Out]

1/86625000*(1312864*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20
-30*x)^(1/2)+1056256*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-2
0-30*x)^(1/2)+9846480*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-
20-30*x)^(1/2)+7921920*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(
-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^3*(-2-2*x)^(1/2)*(-20-30*x)^(1
/2)*(3+2*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^3*(-2-2*x)^(1/2)*(-20-30*x)^
(1/2)*(3+2*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)^(
1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)
^(1/2)*(-20-30*x)^(1/2)[phone]*x^7+33231870*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x
)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x
)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)[phone]*x^6+9969561*15^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1
/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))+8020944*15^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)
*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))[phone]*x^5+8534486800*x^4+10760674300*x^3+7488702560*x^2+2707
141300*x+396786840)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(11/2)

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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((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(13/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.54, size = 166, normalized size = 0.72 \begin {gather*} \frac {338099 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 738486 \, \sqrt {6} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (1312864 \, x^{5} + 11031040 \, x^{4} + 41848650 \, x^{3} + 65139670 \, x^{2} + 42879355 \, x + 9919671\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{155925000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x, algorithm="fricas")

[Out]

1/155925000*(338099*sqrt(6)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*weierstrassPInv
erse(19/27, -28/729, x + 19/18) + 738486*sqrt(6)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x +
729)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18)) + 36*(1312864*x^5 + 1103
1040*x^4 + 41848650*x^3 + 65139670*x^2 + 42879355*x + 9919671)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(64*x^6 +
576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)

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Sympy [F(-1)] Timed out
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((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**(13/2),x)

[Out]

Timed out

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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((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(13/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5)/(2*x + 3)^(13/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{13/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^(13/2),x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^(13/2), x)

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