3.2418 \(\int (5-x) (3+2 x)^4 (2+5 x+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=183 \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^4+\frac{299}{648} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{487}{486} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}}{58320}+\frac{454969 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{559872}-\frac{454969 (6 x+5) \sqrt{3 x^2+5 x+2}}{4478976}+\frac{454969 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{8957952 \sqrt{3}} \]

[Out]

(-454969*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/4478976 + (454969*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/559872 + (487*(
3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/486 + (299*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(5/2))/648 - ((3 + 2*x)^4*(2 + 5*
x + 3*x^2)^(5/2))/27 + ((420721 + 188910*x)*(2 + 5*x + 3*x^2)^(5/2))/58320 + (454969*ArcTanh[(5 + 6*x)/(2*Sqrt
[3]*Sqrt[2 + 5*x + 3*x^2])])/(8957952*Sqrt[3])

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Rubi [A]  time = 0.107975, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \[ -\frac{1}{27} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^4+\frac{299}{648} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac{487}{486} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac{(188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}}{58320}+\frac{454969 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{559872}-\frac{454969 (6 x+5) \sqrt{3 x^2+5 x+2}}{4478976}+\frac{454969 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{8957952 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-454969*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/4478976 + (454969*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/559872 + (487*(
3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2))/486 + (299*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(5/2))/648 - ((3 + 2*x)^4*(2 + 5*
x + 3*x^2)^(5/2))/27 + ((420721 + 188910*x)*(2 + 5*x + 3*x^2)^(5/2))/58320 + (454969*ArcTanh[(5 + 6*x)/(2*Sqrt
[3]*Sqrt[2 + 5*x + 3*x^2])])/(8957952*Sqrt[3])

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*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] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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 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])

Rubi steps

\begin{align*} \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{1}{27} \int (3+2 x)^3 \left (\frac{917}{2}+299 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{1}{648} \int (3+2 x)^2 \left (\frac{36423}{2}+13636 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{\int (3+2 x) \left (\frac{1053773}{2}+396711 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx}{13608}\\ &=\frac{487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac{454969 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{23328}\\ &=\frac{454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac{487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}-\frac{454969 \int \sqrt{2+5 x+3 x^2} \, dx}{373248}\\ &=-\frac{454969 (5+6 x) \sqrt{2+5 x+3 x^2}}{4478976}+\frac{454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac{487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac{454969 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{8957952}\\ &=-\frac{454969 (5+6 x) \sqrt{2+5 x+3 x^2}}{4478976}+\frac{454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac{487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac{454969 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{4478976}\\ &=-\frac{454969 (5+6 x) \sqrt{2+5 x+3 x^2}}{4478976}+\frac{454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac{487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac{299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac{(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac{454969 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{8957952 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0692692, size = 92, normalized size = 0.5 \[ \frac{2274845 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )-6 \sqrt{3 x^2+5 x+2} \left (119439360 x^8+370759680 x^7-2143687680 x^6-14811482880 x^5-37262745216 x^4-49917376080 x^3-37650690888 x^2-15049298650 x-2471988351\right )}{134369280} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-2471988351 - 15049298650*x - 37650690888*x^2 - 49917376080*x^3 - 37262745216*x^4 -
14811482880*x^5 - 2143687680*x^6 + 370759680*x^7 + 119439360*x^8) + 2274845*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt
[6 + 15*x + 9*x^2])])/134369280

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Maple [A]  time = 0.014, size = 149, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{11\,{x}^{3}}{81} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{6133\,{x}^{2}}{486} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{2317\,x}{72} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{2274845+2729814\,x}{559872} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{454969\,\sqrt{3}}{26873856}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{2274845+2729814\,x}{4478976}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{1498291}{58320} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/27*x^4*(3*x^2+5*x+2)^(5/2)+11/81*x^3*(3*x^2+5*x+2)^(5/2)+6133/486*x^2*(3*x^2+5*x+2)^(5/2)+2317/72*x*(3*x^2
+5*x+2)^(5/2)+454969/559872*(5+6*x)*(3*x^2+5*x+2)^(3/2)+454969/26873856*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)
^(1/2))*3^(1/2)-454969/4478976*(5+6*x)*(3*x^2+5*x+2)^(1/2)+1498291/58320*(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 1.51599, size = 225, normalized size = 1.23 \begin{align*} -\frac{16}{27} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{4} + \frac{11}{81} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{3} + \frac{6133}{486} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{2317}{72} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{1498291}{58320} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{454969}{93312} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{2274845}{559872} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{454969}{746496} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{454969}{26873856} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{2274845}{4478976} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/27*(3*x^2 + 5*x + 2)^(5/2)*x^4 + 11/81*(3*x^2 + 5*x + 2)^(5/2)*x^3 + 6133/486*(3*x^2 + 5*x + 2)^(5/2)*x^2
+ 2317/72*(3*x^2 + 5*x + 2)^(5/2)*x + 1498291/58320*(3*x^2 + 5*x + 2)^(5/2) + 454969/93312*(3*x^2 + 5*x + 2)^(
3/2)*x + 2274845/559872*(3*x^2 + 5*x + 2)^(3/2) - 454969/746496*sqrt(3*x^2 + 5*x + 2)*x + 454969/26873856*sqrt
(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 2274845/4478976*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.39545, size = 378, normalized size = 2.07 \begin{align*} -\frac{1}{22394880} \,{\left (119439360 \, x^{8} + 370759680 \, x^{7} - 2143687680 \, x^{6} - 14811482880 \, x^{5} - 37262745216 \, x^{4} - 49917376080 \, x^{3} - 37650690888 \, x^{2} - 15049298650 \, x - 2471988351\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{454969}{53747712} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/22394880*(119439360*x^8 + 370759680*x^7 - 2143687680*x^6 - 14811482880*x^5 - 37262745216*x^4 - 49917376080*
x^3 - 37650690888*x^2 - 15049298650*x - 2471988351)*sqrt(3*x^2 + 5*x + 2) + 454969/53747712*sqrt(3)*log(4*sqrt
(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 4023 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 7938 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 7845 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 3880 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 680 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 128 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 48 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 810 \sqrt{3 x^{2} + 5 x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-4023*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-7938*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-7845
*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3880*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-680*x**5*sqrt(3*
x**2 + 5*x + 2), x) - Integral(128*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(48*x**7*sqrt(3*x**2 + 5*x + 2),
x) - Integral(-810*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]  time = 1.20359, size = 120, normalized size = 0.66 \begin{align*} -\frac{1}{22394880} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (36 \,{\left (2 \,{\left (48 \, x + 149\right )} x - 1723\right )} x - 428573\right )} x - 32346133\right )} x - 346648445\right )} x - 1568778787\right )} x - 7524649325\right )} x - 2471988351\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{454969}{26873856} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/22394880*(2*(12*(6*(8*(30*(36*(2*(48*x + 149)*x - 1723)*x - 428573)*x - 32346133)*x - 346648445)*x - 156877
8787)*x - 7524649325)*x - 2471988351)*sqrt(3*x^2 + 5*x + 2) - 454969/26873856*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt
(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))