3.2452 \(\int \frac{(5-x) (2+5 x+3 x^2)^{7/2}}{3+2 x} \, dx\)

Optimal. Leaf size=169 \[ \frac{1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}-\frac{(7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}}{6912}+\frac{5 (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}}{331776}+\frac{5 (1229315-2568342 x) \sqrt{3 x^2+5 x+2}}{2654208}-\frac{65251715 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{5308416 \sqrt{3}}+\frac{1625}{512} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

(5*(1229315 - 2568342*x)*Sqrt[2 + 5*x + 3*x^2])/2654208 + (5*(6205 - 127338*x)*(2 + 5*x + 3*x^2)^(3/2))/331776
- ((589 + 7446*x)*(2 + 5*x + 3*x^2)^(5/2))/6912 + ((277 - 42*x)*(2 + 5*x + 3*x^2)^(7/2))/672 - (65251715*ArcT
anh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(5308416*Sqrt[3]) + (1625*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[
5]*Sqrt[2 + 5*x + 3*x^2])])/512

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Rubi [A]  time = 0.128263, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {814, 843, 621, 206, 724} \[ \frac{1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}-\frac{(7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}}{6912}+\frac{5 (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}}{331776}+\frac{5 (1229315-2568342 x) \sqrt{3 x^2+5 x+2}}{2654208}-\frac{65251715 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{5308416 \sqrt{3}}+\frac{1625}{512} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(1229315 - 2568342*x)*Sqrt[2 + 5*x + 3*x^2])/2654208 + (5*(6205 - 127338*x)*(2 + 5*x + 3*x^2)^(3/2))/331776
- ((589 + 7446*x)*(2 + 5*x + 3*x^2)^(5/2))/6912 + ((277 - 42*x)*(2 + 5*x + 3*x^2)^(7/2))/672 - (65251715*ArcT
anh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(5308416*Sqrt[3]) + (1625*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[
5]*Sqrt[2 + 5*x + 3*x^2])])/512

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 621

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

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]

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{3+2 x} \, dx &=\frac{1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{1}{192} \int \frac{(2163+2482 x) \left (2+5 x+3 x^2\right )^{5/2}}{3+2 x} \, dx\\ &=-\frac{(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac{1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac{\int \frac{(-355890-424460 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx}{27648}\\ &=\frac{5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac{(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac{1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{\int \frac{(43354260+51366840 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx}{2654208}\\ &=\frac{5 (1229315-2568342 x) \sqrt{2+5 x+3 x^2}}{2654208}+\frac{5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac{(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac{1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac{\int \frac{-2676363480-3132082320 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{127401984}\\ &=\frac{5 (1229315-2568342 x) \sqrt{2+5 x+3 x^2}}{2654208}+\frac{5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac{(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac{1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{65251715 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{5308416}+\frac{8125}{512} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{5 (1229315-2568342 x) \sqrt{2+5 x+3 x^2}}{2654208}+\frac{5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac{(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac{1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{65251715 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{2654208}-\frac{8125}{256} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{5 (1229315-2568342 x) \sqrt{2+5 x+3 x^2}}{2654208}+\frac{5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac{(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac{1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac{65251715 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{5308416 \sqrt{3}}+\frac{1625}{512} \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0914347, size = 123, normalized size = 0.73 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (31352832 x^7-50015232 x^6-529784064 x^5-1167854976 x^4-1224844848 x^3-722869752 x^2-185981750 x-101435865\right )-353808000 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-456762005 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{111476736} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-101435865 - 185981750*x - 722869752*x^2 - 1224844848*x^3 - 1167854976*x^4 - 529784
064*x^5 - 50015232*x^6 + 31352832*x^7) - 353808000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)] - 456762005*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/111476736

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Maple [B]  time = 0.006, size = 295, normalized size = 1.8 \begin{align*} -{\frac{5+6\,x}{96} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{35+42\,x}{6912} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{175+210\,x}{331776} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{175+210\,x}{2654208}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{35\,\sqrt{3}}{15925248}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{28} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{65+78\,x}{72} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{5525+6630\,x}{3456} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{111475+133770\,x}{27648}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{679705\,\sqrt{3}}{165888}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{13}{16} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{325}{192} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{1625}{512}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{1625\,\sqrt{5}}{512}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(5+6*x)*(3*x^2+5*x+2)^(7/2)+7/6912*(5+6*x)*(3*x^2+5*x+2)^(5/2)-35/331776*(5+6*x)*(3*x^2+5*x+2)^(3/2)+35/
2654208*(5+6*x)*(3*x^2+5*x+2)^(1/2)-35/15925248*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+13/28*(3
*(x+3/2)^2-4*x-19/4)^(7/2)-13/72*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1105/3456*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)
^(3/2)-22295/27648*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-679705/165888*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*
x-19/4)^(1/2))*3^(1/2)+13/16*(3*(x+3/2)^2-4*x-19/4)^(5/2)+325/192*(3*(x+3/2)^2-4*x-19/4)^(3/2)+1625/512*(12*(x
+3/2)^2-16*x-19)^(1/2)-1625/512*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 1.80572, size = 251, normalized size = 1.49 \begin{align*} -\frac{1}{16} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{277}{672} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{1241}{1152} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{589}{6912} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{106115}{55296} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{31025}{331776} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{2140285}{442368} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{65251715}{15925248} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{1625}{512} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{6146575}{2654208} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(3*x^2 + 5*x + 2)^(7/2)*x + 277/672*(3*x^2 + 5*x + 2)^(7/2) - 1241/1152*(3*x^2 + 5*x + 2)^(5/2)*x - 589/
6912*(3*x^2 + 5*x + 2)^(5/2) - 106115/55296*(3*x^2 + 5*x + 2)^(3/2)*x + 31025/331776*(3*x^2 + 5*x + 2)^(3/2) -
2140285/442368*sqrt(3*x^2 + 5*x + 2)*x - 65251715/15925248*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x +
5/2) - 1625/512*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 6146575/26542
08*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.48089, size = 494, normalized size = 2.92 \begin{align*} -\frac{1}{18579456} \,{\left (31352832 \, x^{7} - 50015232 \, x^{6} - 529784064 \, x^{5} - 1167854976 \, x^{4} - 1224844848 \, x^{3} - 722869752 \, x^{2} - 185981750 \, x - 101435865\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{65251715}{31850496} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + \frac{1625}{1024} \, \sqrt{5} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18579456*(31352832*x^7 - 50015232*x^6 - 529784064*x^5 - 1167854976*x^4 - 1224844848*x^3 - 722869752*x^2 - 1
85981750*x - 101435865)*sqrt(3*x^2 + 5*x + 2) + 65251715/31850496*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)
*(6*x + 5) + 72*x^2 + 120*x + 49) + 1625/1024*sqrt(5)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2
+ 212*x + 89)/(4*x^2 + 12*x + 9))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.29197, size = 211, normalized size = 1.25 \begin{align*} -\frac{1}{18579456} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (42 \, x - 67\right )} x - 25549\right )} x - 337921\right )} x - 2835289\right )} x - 30119573\right )} x - 92990875\right )} x - 101435865\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{1625}{512} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{65251715}{15925248} \, \sqrt{3} \log \left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18579456*(2*(12*(18*(8*(6*(36*(42*x - 67)*x - 25549)*x - 337921)*x - 2835289)*x - 30119573)*x - 92990875)*x
- 101435865)*sqrt(3*x^2 + 5*x + 2) + 1625/512*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3
*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 65251715/15925248*sqrt
(3)*log(abs(-6*sqrt(3)*x - 5*sqrt(3) + 6*sqrt(3*x^2 + 5*x + 2)))