3.66 \(\int (3-x+2 x^2)^{3/2} (2+3 x+5 x^2)^3 \, dx\)

Optimal. Leaf size=189 \[ \frac{25}{4} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{725}{48} \left (2 x^2-x+3\right )^{5/2} x^4+\frac{27785 \left (2 x^2-x+3\right )^{5/2} x^3}{1536}+\frac{384739 \left (2 x^2-x+3\right )^{5/2} x^2}{43008}-\frac{81685 \left (2 x^2-x+3\right )^{5/2} x}{114688}-\frac{4625907 \left (2 x^2-x+3\right )^{5/2}}{2293760}-\frac{667795 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2097152}-\frac{46077855 (1-4 x) \sqrt{2 x^2-x+3}}{33554432}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}} \]

[Out]

(-46077855*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/33554432 - (667795*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/2097152 - (46259
07*(3 - x + 2*x^2)^(5/2))/2293760 - (81685*x*(3 - x + 2*x^2)^(5/2))/114688 + (384739*x^2*(3 - x + 2*x^2)^(5/2)
)/43008 + (27785*x^3*(3 - x + 2*x^2)^(5/2))/1536 + (725*x^4*(3 - x + 2*x^2)^(5/2))/48 + (25*x^5*(3 - x + 2*x^2
)^(5/2))/4 - (1059790665*ArcSinh[(1 - 4*x)/Sqrt[23]])/(67108864*Sqrt[2])

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Rubi [A]  time = 0.18971, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac{25}{4} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{725}{48} \left (2 x^2-x+3\right )^{5/2} x^4+\frac{27785 \left (2 x^2-x+3\right )^{5/2} x^3}{1536}+\frac{384739 \left (2 x^2-x+3\right )^{5/2} x^2}{43008}-\frac{81685 \left (2 x^2-x+3\right )^{5/2} x}{114688}-\frac{4625907 \left (2 x^2-x+3\right )^{5/2}}{2293760}-\frac{667795 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2097152}-\frac{46077855 (1-4 x) \sqrt{2 x^2-x+3}}{33554432}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-46077855*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/33554432 - (667795*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/2097152 - (46259
07*(3 - x + 2*x^2)^(5/2))/2293760 - (81685*x*(3 - x + 2*x^2)^(5/2))/114688 + (384739*x^2*(3 - x + 2*x^2)^(5/2)
)/43008 + (27785*x^3*(3 - x + 2*x^2)^(5/2))/1536 + (725*x^4*(3 - x + 2*x^2)^(5/2))/48 + (25*x^5*(3 - x + 2*x^2
)^(5/2))/4 - (1059790665*ArcSinh[(1 - 4*x)/Sqrt[23]])/(67108864*Sqrt[2])

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx &=\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{20} \int \left (3-x+2 x^2\right )^{3/2} \left (160+720 x+2280 x^2+4140 x^3+3825 x^4+\frac{10875 x^5}{2}\right ) \, dx\\ &=\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{360} \int \left (3-x+2 x^2\right )^{3/2} \left (2880+12960 x+41040 x^2+9270 x^3+\frac{416775 x^4}{4}\right ) \, dx\\ &=\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (3-x+2 x^2\right )^{3/2} \left (46080+207360 x-\frac{1124415 x^2}{4}+\frac{5771085 x^3}{8}\right ) \, dx}{5760}\\ &=\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (645120-\frac{5701095 x}{4}-\frac{11027475 x^2}{16}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{80640}\\ &=-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (\frac{156945465}{16}-\frac{624497445 x}{32}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{967680}\\ &=-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{667795 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{131072}\\ &=-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{46077855 \int \sqrt{3-x+2 x^2} \, dx}{4194304}\\ &=-\frac{46077855 (1-4 x) \sqrt{3-x+2 x^2}}{33554432}-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{1059790665 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{67108864}\\ &=-\frac{46077855 (1-4 x) \sqrt{3-x+2 x^2}}{33554432}-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\left (46077855 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{67108864}\\ &=-\frac{46077855 (1-4 x) \sqrt{3-x+2 x^2}}{33554432}-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.2339, size = 85, normalized size = 0.45 \[ \frac{4 \sqrt{2 x^2-x+3} \left (88080384000 x^9+124780544000 x^8+328328806400 x^7+430820229120 x^6+571298324480 x^5+487891884032 x^4+389257196928 x^3+199615064544 x^2+53985432012 x-72152399943\right )-111278019825 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{14092861440} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-72152399943 + 53985432012*x + 199615064544*x^2 + 389257196928*x^3 + 487891884032*x^4
+ 571298324480*x^5 + 430820229120*x^6 + 328328806400*x^7 + 124780544000*x^8 + 88080384000*x^9) - 111278019825*
Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/14092861440

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Maple [A]  time = 0.059, size = 151, normalized size = 0.8 \begin{align*}{\frac{25\,{x}^{5}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{725\,{x}^{4}}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{27785\,{x}^{3}}{1536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{384739\,{x}^{2}}{43008} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{81685\,x}{114688} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-46077855+184311420\,x}{33554432}\sqrt{2\,{x}^{2}-x+3}}+{\frac{1059790665\,\sqrt{2}}{134217728}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-667795+2671180\,x}{2097152} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{4625907}{2293760} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}

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)

[Out]

25/4*x^5*(2*x^2-x+3)^(5/2)+725/48*x^4*(2*x^2-x+3)^(5/2)+27785/1536*x^3*(2*x^2-x+3)^(5/2)+384739/43008*x^2*(2*x
^2-x+3)^(5/2)-81685/114688*x*(2*x^2-x+3)^(5/2)[phone]/33554432*(-1+4*x)*(2*x^2-x+3)^(1/2)[phone]/1342177
28*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+667795/2097152*(-1+4*x)*(2*x^2-x+3)^(3/2)-4625907/2293760*(2*x^2-x+3
)^(5/2)

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Maxima [A]  time = 1.50344, size = 232, normalized size = 1.23 \begin{align*} \frac{25}{4} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{5} + \frac{725}{48} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{4} + \frac{27785}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{384739}{43008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{81685}{114688} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{4625907}{2293760} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{667795}{524288} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{667795}{2097152} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{46077855}{8388608} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{1059790665}{134217728} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{46077855}{33554432} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

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]

25/4*(2*x^2 - x + 3)^(5/2)*x^5 + 725/48*(2*x^2 - x + 3)^(5/2)*x^4 + 27785/1536*(2*x^2 - x + 3)^(5/2)*x^3 + 384
739/43008*(2*x^2 - x + 3)^(5/2)*x^2 - 81685/114688*(2*x^2 - x + 3)^(5/2)*x - 4625907/2293760*(2*x^2 - x + 3)^(
5/2) + 667795/524288*(2*x^2 - x + 3)^(3/2)*x - 667795/2097152*(2*x^2 - x + 3)^(3/2) + 46077855/8388608*sqrt(2*
x^2 - x + 3)*x + 1059790665/134217728*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 46077855/33554432*sqrt(2*x^2
- x + 3)

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Fricas [A]  time = 1.59326, size = 423, normalized size = 2.24 \begin{align*} \frac{1}{3523215360} \,{\left (88080384000 \, x^{9} + 124780544000 \, x^{8} + 328328806400 \, x^{7} + 430820229120 \, x^{6} + 571298324480 \, x^{5} + 487891884032 \, x^{4} + 389257196928 \, x^{3} + 199615064544 \, x^{2} + 53985432012 \, x - 72152399943\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{1059790665}{268435456} \, \sqrt{2} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end{align*}

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]

1/3523215360*(88080384000*x^9 + 124780544000*x^8 + 328328806400*x^7 + 430820229120*x^6 + 571298324480*x^5 + 48
7891884032*x^4 + 389257196928*x^3 + 199615064544*x^2 + 53985432012*x - 72152399943)*sqrt(2*x^2 - x + 3) + 1059
790665/268435456*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \end{align*}

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 [A]  time = 1.16042, size = 126, normalized size = 0.67 \begin{align*} \frac{1}{3523215360} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (8 \,{\left (140 \,{\left (160 \,{\left (12 \, x + 17\right )} x + 7157\right )} x + 1314759\right )} x + 13947713\right )} x + 238228459\right )} x + 3041071851\right )} x + 6237970767\right )} x + 13496358003\right )} x - 72152399943\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{1059790665}{134217728} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}

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]

1/3523215360*(4*(8*(4*(16*(20*(8*(140*(160*(12*x + 17)*x + 7157)*x + 1314759)*x + 13947713)*x + 238228459)*x +
3041071851)*x + 6237970767)*x + 13496358003)*x - 72152399943)*sqrt(2*x^2 - x + 3) - 1059790665/134217728*sqrt
(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)