3.1.71 \(\int \frac {(3-x+2 x^2)^{3/2}}{(2+3 x+5 x^2)^3} \, dx\) [71]
Optimal. Leaf size=223 \[ \frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 (277+696 x) \sqrt {3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (29367+20575 \sqrt {2}+\left (70517+49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688}-\frac {3 \sqrt {\frac {1}{682} \left (-366990269+259509026 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-366990269+259509026 \sqrt {2}\right )}} \left (29367-20575 \sqrt {2}+\left (70517-49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688} \]
[Out]
1/62*(3+10*x)*(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^2+3/3844*(277+696*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)-3/5243216*a
rctanh(1/31*(29367+x*(70517-49942*2^(1/2))-20575*2^(1/2))*341^(1/2)/(-366990269+259509026*2^(1/2))^(1/2)/(2*x^
2-x+3)^(1/2))*(-250287363458+176985155732*2^(1/2))^(1/2)+3/5243216*arctan(1/31*(29367+20575*2^(1/2)+x*(70517+4
9942*2^(1/2)))*341^(1/2)/(366990269+259509026*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(250287363458+176985155732*2^(
1/2))^(1/2)
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Rubi [A]
time = 0.28, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {985, 1027,
1049, 1043, 212, 210} \begin {gather*} \frac {3 \sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )}{7688}+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac {3 (696 x+277) \sqrt {2 x^2-x+3}}{3844 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{682} \left (259509026 \sqrt {2}-366990269\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (259509026 \sqrt {2}-366990269\right )}} \left (\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )}{7688} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*(277 + 696*x)*Sqrt[3 - x + 2*x^2])/(3844*(2 +
3*x + 5*x^2)) + (3*Sqrt[(366990269 + 259509026*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(366990269 + 259509026*Sqrt[
2]))]*(29367 + 20575*Sqrt[2] + (70517 + 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688 - (3*Sqrt[(-366990269 +
259509026*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-366990269 + 259509026*Sqrt[2]))]*(29367 - 20575*Sqrt[2] + (7051
7 - 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 985
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b +
2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Dist[1/((b^2 - 4*a*c)*(p
+ 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e
*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
Rule 1027
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[(g*b - 2*a*h - (b*h - 2*g*c)*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/((b^2 - 4*a*c)*(p
+ 1))), x] - Dist[1/((b^2 - 4*a*c)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q
*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h
- 2*g*c)*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0]
Rule 1043
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]
Rule 1049
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {1}{62} \int \frac {\left (-\frac {189}{2}+33 x\right ) \sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 (277+696 x) \sqrt {3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {\frac {13359}{4}-1353 x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{1922}\\ &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 (277+696 x) \sqrt {3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {-\frac {33}{4} \left (6257-4453 \sqrt {2}\right )+\frac {33}{4} \left (2649-1804 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{42284 \sqrt {2}}-\frac {\int \frac {-\frac {33}{4} \left (6257+4453 \sqrt {2}\right )+\frac {33}{4} \left (2649+1804 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{42284 \sqrt {2}}\\ &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 (277+696 x) \sqrt {3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {\left (99 \left (519018052-366990269 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {33759}{16} \left (366990269-259509026 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {-\frac {33}{4} \left (29367-20575 \sqrt {2}\right )-\frac {33}{4} \left (70517-49942 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{61504}+\frac {\left (99 \left (519018052+366990269 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {33759}{16} \left (366990269+259509026 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {-\frac {33}{4} \left (29367+20575 \sqrt {2}\right )-\frac {33}{4} \left (70517+49942 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{61504}\\ &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 (277+696 x) \sqrt {3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (29367+20575 \sqrt {2}+\left (70517+49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688}-\frac {3 \sqrt {\frac {1}{682} \left (-366990269+259509026 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-366990269+259509026 \sqrt {2}\right )}} \left (29367-20575 \sqrt {2}+\left (70517-49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.79, size = 572, normalized size = 2.57 \begin {gather*} \frac {\frac {3306250 \sqrt {3-x+2 x^2} \left (2220+8343 x+10171 x^2+11680 x^3\right )}{\left (2+3 x+5 x^2\right )^2}-42578694225 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+406695200 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {93 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+10 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+14 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {4926449381 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-2660991465 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-186 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {155209944 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-248390285 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{12709225000} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
((3306250*Sqrt[3 - x + 2*x^2]*(2220 + 8343*x + 10171*x^2 + 11680*x^3))/(2 + 3*x + 5*x^2)^2 - 42578694225*RootS
um[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]/(-
13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 406695200*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[
2]*#1^3 - 5*#1^4 & , (93*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 10*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 -
x + 2*x^2] - #1]*#1)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 14*RootSum[-56 - 26*Sqrt[2]*#1 +
17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (4926449381*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 26
60991465*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) &
] - 186*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (155209944*Sqrt[2]*Log[-(Sqrt[2]*
x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 248390285*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2]
+ 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ])/12709225000
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(81551\) vs.
\(2(171)=342\).
time = 0.97, size = 81552, normalized size = 365.70 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
[Out]
result too large to display
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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*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
[Out]
integrate((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2)^3, x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2183 vs.
\(2 (171) = 342\).
time = 3.27, size = 2183, normalized size = 9.79 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
[Out]
-1/85773071417697924109696*(189113268*134689869150937352^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(2)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4)*sqrt(366990269*sqrt(2) + 519018052)*arctan(1/1067259092343193675559267622545473*(1608
9559612*sqrt(129754513)*(11*134689869150937352^(3/4)*sqrt(341)*(38305160*x^7 - 147261352*x^6 + 309398878*x^5 -
495410374*x^4 + 248212864*x^3 - 117285552*x^2 - sqrt(2)*(26988622*x^7 - 104036813*x^6 + 218448200*x^5 - 35057
9241*x^4 + 175844824*x^3 - 83534472*x^2 - 191303424*x + 135585792) - 271171584*x + 191303424) + 4022389903*134
689869150937352^(1/4)*sqrt(341)*(2906601*x^7 - 44604657*x^6 + 235604928*x^5 - 537156764*x^4 + 693706464*x^3 -
436717728*x^2 - sqrt(2)*(2050114*x^7 - 31475955*x^6 + 166375268*x^5 - 379661892*x^4 + 490500864*x^3 - 30982780
8*x^2 - 348696576*x + 246965760) - 493931520*x + 348696576))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(2) + 5190
18052) + 3029638713748420756426308089806504*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^
5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*
x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(259509026/713)*(sqrt(1297545
13)*(11*134689869150937352^(3/4)*sqrt(341)*(5980372*x^7 - 8582986*x^6 + 27618126*x^5 - 10751392*x^4 + 12649968
*x^3 + 12517632*x^2 - sqrt(2)*(4201650*x^7 - 6032009*x^6 + 19421619*x^5 - 7633552*x^4 + 9050328*x^3 + 8640000*
x^2 - 8640000*x) - 12517632*x) + 4022389903*134689869150937352^(1/4)*sqrt(341)*(453599*x^7 - 5867420*x^6 + 226
22900*x^5 - 29282112*x^4 + 37610208*x^3 + 22726656*x^2 - sqrt(2)*(319303*x^7 - 4130364*x^6 + 15927060*x^5 - 20
630592*x^4 + 26556768*x^3 + 15800832*x^2 - 15800832*x) - 22726656*x))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(
2) + 519018052) + 8186887989068712800954*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5
+ 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960
*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) + 372131272230396036407*sqrt(31)*(254591*x^8 - 4815
126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 -
76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt(-(1346898691509373
52^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(696*x + 277) - 973*x - 419)*sqrt(366
990269*sqrt(2) + 519018052) - 4356437317274441*x^2 - 3911902897144396*sqrt(2)*(2*x^2 - x + 3) + 13424939487927
359*x - 17781376805201800)/x^2) + 34427712656232054050298955565983*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385
560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9
789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8
- 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 1857945
6)) + 189113268*134689869150937352^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x +
4)*sqrt(366990269*sqrt(2) + 519018052)*arctan(1/1067259092343193675559267622545473*(16089559612*sqrt(129754513
)*(11*134689869150937352^(3/4)*sqrt(341)*(38305160*x^7 - 147261352*x^6 + 309398878*x^5 - 495410374*x^4 + 24821
2864*x^3 - 117285552*x^2 - sqrt(2)*(26988622*x^7 - 104036813*x^6 + 218448200*x^5 - 350579241*x^4 + 175844824*x
^3 - 83534472*x^2 - 191303424*x + 135585792) - 271171584*x + 191303424) + 4022389903*134689869150937352^(1/4)*
sqrt(341)*(2906601*x^7 - 44604657*x^6 + 235604928*x^5 - 537156764*x^4 + 693706464*x^3 - 436717728*x^2 - sqrt(2
)*(2050114*x^7 - 31475955*x^6 + 166375268*x^5 - 379661892*x^4 + 490500864*x^3 - 309827808*x^2 - 348696576*x +
246965760) - 493931520*x + 348696576))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(2) + 519018052) - 3029638713748
420756426308089806504*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 6420
48*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396
144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(259509026/713)*(sqrt(129754513)*(11*13468986915093
7352^(3/4)*sqrt(341)*(5980372*x^7 - 8582986*x^6 + 27618126*x^5 - 10751392*x^4 + 12649968*x^3 + 12517632*x^2 -
sqrt(2)*(4201650*x^7 - 6032009*x^6 + 19421619*x^5 - 7633552*x^4 + 9050328*x^3 + 8640000*x^2 - 8640000*x) - 125
17632*x) + 4022389903*134689869150937352^(1/4)*sqrt(341)*(453599*x^7 - 5867420*x^6 + 22622900*x^5 - 29282112*x
^4 + 37610208*x^3 + 22726656*x^2 - sqrt(2)*(319303*x^7 - 4130364*x^6 + 15927060*x^5 - 20630592*x^4 + 26556768*
x^3 + 15800832*x^2 - 15800832*x) - 22726656*x))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(2) + 519018052) - 8186
887989068712800954*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336
*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1
209600*x^2 - 1036800*x) + 3276288*x) - 37213127...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x**2-x+3)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral((2*x**2 - x + 3)**(3/2)/(5*x**2 + 3*x + 2)**3, x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Francis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,i
nfinity,inf
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (2\,x^2-x+3\right )}^{3/2}}{{\left (5\,x^2+3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^2 - x + 3)^(3/2)/(3*x + 5*x^2 + 2)^3,x)
[Out]
int((2*x^2 - x + 3)^(3/2)/(3*x + 5*x^2 + 2)^3, x)
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