### 3.339 $$\int \frac{(3-x+2 x^2)^{3/2} (2+x+3 x^2-x^3+5 x^4)}{(5+2 x)^4} \, dx$$

Optimal. Leaf size=181 $-\frac{32865365 \left (2 x^2-x+3\right )^{5/2}}{17915904 (2 x+5)}+\frac{556255 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^2}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}-\frac{(138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}}{17915904}-\frac{(135068604-22512089 x) \sqrt{2 x^2-x+3}}{331776}+\frac{517762327 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{221184 \sqrt{2}}-\frac{19176431 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}}$

[Out]

-((135068604 - 22512089*x)*Sqrt[3 - x + 2*x^2])/331776 - ((138006843 - 34265045*x)*(3 - x + 2*x^2)^(3/2))/1791
5904 - (3667*(3 - x + 2*x^2)^(5/2))/(1728*(5 + 2*x)^3) + (556255*(3 - x + 2*x^2)^(5/2))/(248832*(5 + 2*x)^2) -
(32865365*(3 - x + 2*x^2)^(5/2))/(17915904*(5 + 2*x)) - (19176431*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8192*Sqrt[2])
+ (517762327*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(221184*Sqrt[2])

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Rubi [A]  time = 0.267145, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 40, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.175, Rules used = {1650, 814, 843, 619, 215, 724, 206} $-\frac{32865365 \left (2 x^2-x+3\right )^{5/2}}{17915904 (2 x+5)}+\frac{556255 \left (2 x^2-x+3\right )^{5/2}}{248832 (2 x+5)^2}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{1728 (2 x+5)^3}-\frac{(138006843-34265045 x) \left (2 x^2-x+3\right )^{3/2}}{17915904}-\frac{(135068604-22512089 x) \sqrt{2 x^2-x+3}}{331776}+\frac{517762327 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{221184 \sqrt{2}}-\frac{19176431 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^4,x]

[Out]

-((135068604 - 22512089*x)*Sqrt[3 - x + 2*x^2])/331776 - ((138006843 - 34265045*x)*(3 - x + 2*x^2)^(3/2))/1791
5904 - (3667*(3 - x + 2*x^2)^(5/2))/(1728*(5 + 2*x)^3) + (556255*(3 - x + 2*x^2)^(5/2))/(248832*(5 + 2*x)^2) -
(32865365*(3 - x + 2*x^2)^(5/2))/(17915904*(5 + 2*x)) - (19176431*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8192*Sqrt[2])
+ (517762327*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(221184*Sqrt[2])

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 814

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 843

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]

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]

Rule 724

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^4} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}-\frac{1}{216} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{43355}{16}-\frac{11605 x}{2}+1458 x^2-540 x^3\right )}{(5+2 x)^3} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{4202675}{16}-\frac{2477469 x}{4}+38880 x^2\right )}{(5+2 x)^2} \, dx}{31104}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}-\frac{\int \frac{\left (\frac{182685181}{16}-34265045 x\right ) \left (3-x+2 x^2\right )^{3/2}}{5+2 x} \, dx}{2239488}\\ &=-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}+\frac{\int \frac{(-14584438152+38900889792 x) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{143327232}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}-\frac{\int \frac{10736183791872-21472693553664 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{4586471424}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}+\frac{19176431 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{8192}-\frac{517762327 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{36864}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}+\frac{517762327 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{18432}+\frac{19176431 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{8192 \sqrt{46}}\\ &=-\frac{(135068604-22512089 x) \sqrt{3-x+2 x^2}}{331776}-\frac{(138006843-34265045 x) \left (3-x+2 x^2\right )^{3/2}}{17915904}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{1728 (5+2 x)^3}+\frac{556255 \left (3-x+2 x^2\right )^{5/2}}{248832 (5+2 x)^2}-\frac{32865365 \left (3-x+2 x^2\right )^{5/2}}{17915904 (5+2 x)}-\frac{19176431 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}}+\frac{517762327 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{221184 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.220023, size = 108, normalized size = 0.6 $\frac{\frac{12 \sqrt{2 x^2-x+3} \left (46080 x^6-315648 x^5+2626848 x^4-33595416 x^3-594798908 x^2-2006873194 x-1994650739\right )}{(2 x+5)^3}+517762327 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-517763637 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{442368}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((3 - x + 2*x^2)^(3/2)*(2 + x + 3*x^2 - x^3 + 5*x^4))/(5 + 2*x)^4,x]

[Out]

((12*Sqrt[3 - x + 2*x^2]*(-1994650739 - 2006873194*x - 594798908*x^2 - 33595416*x^3 + 2626848*x^4 - 315648*x^5
+ 46080*x^6))/(5 + 2*x)^3 - 517763637*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 517762327*Sqrt[2]*ArcTanh[(17 - 2
2*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/442368

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Maple [A]  time = 0.062, size = 221, normalized size = 1.2 \begin{align*}{\frac{-22400309+89601236\,x}{1327104}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{556255}{995328} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{517762327\,\sqrt{2}}{442368}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }+{\frac{-345+1380\,x}{4096}\sqrt{2\,{x}^{2}-x+3}}+{\frac{19176431\,\sqrt{2}}{16384}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-5+20\,x}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{517762327}{71663616} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{517762327}{1327104}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{3667}{13824} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}+{\frac{-32865365+131461460\,x}{71663616} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}-{\frac{32865365}{35831808} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x)

[Out]

22400309/1327104*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(1/2)+556255/995328/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(5/2)+
517762327/442368*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))+345/4096*(-1+4*x)*(2*
x^2-x+3)^(1/2)+19176431/16384*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+5/256*(-1+4*x)*(2*x^2-x+3)^(3/2)-51776232
7/71663616*(2*(x+5/2)^2-11*x-19/2)^(3/2)-517762327/1327104*(2*(x+5/2)^2-11*x-19/2)^(1/2)-3667/13824/(x+5/2)^3*
(2*(x+5/2)^2-11*x-19/2)^(5/2)+32865365/71663616*(-1+4*x)*(2*(x+5/2)^2-11*x-19/2)^(3/2)-32865365/35831808/(x+5/
2)*(2*(x+5/2)^2-11*x-19/2)^(5/2)

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Maxima [A]  time = 1.5567, size = 255, normalized size = 1.41 \begin{align*} \frac{5}{64} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{1094743}{497664} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1728 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac{556255 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{248832 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{22512089}{331776} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{19176431}{16384} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{517762327}{442368} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) - \frac{11255717}{27648} \, \sqrt{2 \, x^{2} - x + 3} - \frac{32865365 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{995328 \,{\left (2 \, x + 5\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x, algorithm="maxima")

[Out]

5/64*(2*x^2 - x + 3)^(3/2)*x - 1094743/497664*(2*x^2 - x + 3)^(3/2) - 3667/1728*(2*x^2 - x + 3)^(5/2)/(8*x^3 +
60*x^2 + 150*x + 125) + 556255/248832*(2*x^2 - x + 3)^(5/2)/(4*x^2 + 20*x + 25) + 22512089/331776*sqrt(2*x^2
- x + 3)*x + 19176431/16384*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 517762327/442368*sqrt(2)*arcsin
h(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) - 11255717/27648*sqrt(2*x^2 - x + 3) - 32865365
/995328*(2*x^2 - x + 3)^(3/2)/(2*x + 5)

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Fricas [A]  time = 1.48414, size = 581, normalized size = 3.21 \begin{align*} \frac{517763637 \, \sqrt{2}{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 517762327 \, \sqrt{2}{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )} \log \left (\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 24 \,{\left (46080 \, x^{6} - 315648 \, x^{5} + 2626848 \, x^{4} - 33595416 \, x^{3} - 594798908 \, x^{2} - 2006873194 \, x - 1994650739\right )} \sqrt{2 \, x^{2} - x + 3}}{884736 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x, algorithm="fricas")

[Out]

1/884736*(517763637*sqrt(2)*(8*x^3 + 60*x^2 + 150*x + 125)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x
^2 + 16*x - 25) + 517762327*sqrt(2)*(8*x^3 + 60*x^2 + 150*x + 125)*log((24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x -
17) - 1060*x^2 + 1036*x - 1153)/(4*x^2 + 20*x + 25)) + 24*(46080*x^6 - 315648*x^5 + 2626848*x^4 - 33595416*x^
3 - 594798908*x^2 - 2006873194*x - 1994650739)*sqrt(2*x^2 - x + 3))/(8*x^3 + 60*x^2 + 150*x + 125)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-x+3)**(3/2)*(5*x**4-x**3+3*x**2+x+2)/(5+2*x)**4,x)

[Out]

Integral((2*x**2 - x + 3)**(3/2)*(5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x + 5)**4, x)

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Giac [B]  time = 1.24558, size = 424, normalized size = 2.34 \begin{align*} \frac{1}{4096} \,{\left (4 \,{\left (8 \,{\left (20 \, x - 287\right )} x + 23341\right )} x - 1004633\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{19176431}{16384} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{517762327}{442368} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{517762327}{442368} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{\sqrt{2}{\left (1092794276 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} + 18284336132 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} + 20314214356 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 151449344092 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 102529692109 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 41882448755\right )}}{36864 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^4-x^3+3*x^2+x+2)/(5+2*x)^4,x, algorithm="giac")

[Out]

1/4096*(4*(8*(20*x - 287)*x + 23341)*x - 1004633)*sqrt(2*x^2 - x + 3) - 19176431/16384*sqrt(2)*log(-2*sqrt(2)*
(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 517762327/442368*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^
2 - x + 3))) - 517762327/442368*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/36864*
sqrt(2)*(1092794276*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 18284336132*(sqrt(2)*x - sqrt(2*x^2 - x + 3)
)^4 + 20314214356*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 - 151449344092*(sqrt(2)*x - sqrt(2*x^2 - x + 3))
^2 + 102529692109*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 41882448755)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3)
)^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^3