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

Optimal. Leaf size=133 \[ -\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac {(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac {1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {1129 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {1129 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \]

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Rubi [A]  time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \begin {gather*} -\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac {(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac {1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {1129 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {1129 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1129*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/20736 + (1129*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/2592 - ((3 + 2*x)^2*(
2 + 5*x + 3*x^2)^(5/2))/21 + ((5827 + 2370*x)*(2 + 5*x + 3*x^2)^(5/2))/1890 + (1129*ArcTanh[(5 + 6*x)/(2*Sqrt[
3]*Sqrt[2 + 5*x + 3*x^2])])/(41472*Sqrt[3])

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 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 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 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])

Rubi steps

\begin {align*} \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {1}{21} \int (3+2 x) \left (\frac {721}{2}+237 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129}{108} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}-\frac {1129 \int \sqrt {2+5 x+3 x^2} \, dx}{1728}\\ &=-\frac {1129 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{41472}\\ &=-\frac {1129 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{20736}\\ &=-\frac {1129 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{41472 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 82, normalized size = 0.62 \begin {gather*} \frac {39515 \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 (1244160 x^6-311040 x^5-27084672 x^4-79049520 x^3-94861176 x^2-51971350 x-10669737\right )}{4354560} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-10669737 - 51971350*x - 94861176*x^2 - 79049520*x^3 - 27084672*x^4 - 311040*x^5 +
1244160*x^6) + 39515*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/4354560

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IntegrateAlgebraic [A]  time = 0.71, size = 84, normalized size = 0.63 \begin {gather*} \frac {1129 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{20736 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-1244160 x^6+311040 x^5+27084672 x^4+79049520 x^3+94861176 x^2+51971350 x+10669737\right )}{725760} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(10669737 + 51971350*x + 94861176*x^2 + 79049520*x^3 + 27084672*x^4 + 311040*x^5 - 1244
160*x^6))/725760 + (1129*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(20736*Sqrt[3])

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fricas [A]  time = 0.41, size = 83, normalized size = 0.62 \begin {gather*} -\frac {1}{725760} \, {\left (1244160 \, x^{6} - 311040 \, x^{5} - 27084672 \, x^{4} - 79049520 \, x^{3} - 94861176 \, x^{2} - 51971350 \, x - 10669737\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1129}{248832} \, \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/725760*(1244160*x^6 - 311040*x^5 - 27084672*x^4 - 79049520*x^3 - 94861176*x^2 - 51971350*x - 10669737)*sqrt
(3*x^2 + 5*x + 2) + 1129/248832*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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giac [A]  time = 0.21, size = 79, normalized size = 0.59 \begin {gather*} -\frac {1}{725760} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (90 \, {\left (4 \, x - 1\right )} x - 7837\right )} x - 182985\right )} x - 3952549\right )} x - 25985675\right )} x - 10669737\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1129}{124416} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/725760*(2*(12*(18*(8*(90*(4*x - 1)*x - 7837)*x - 182985)*x - 3952549)*x - 25985675)*x - 10669737)*sqrt(3*x^
2 + 5*x + 2) - 1129/124416*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))

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maple [A]  time = 0.05, size = 115, normalized size = 0.86 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x^{2}}{21}+\frac {43 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x}{63}+\frac {1129 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{124416}+\frac {5017 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1890}+\frac {1129 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{2592}-\frac {1129 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{20736} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/21*(3*x^2+5*x+2)^(5/2)*x^2+43/63*(3*x^2+5*x+2)^(5/2)*x+5017/1890*(3*x^2+5*x+2)^(5/2)+1129/2592*(6*x+5)*(3*x
^2+5*x+2)^(3/2)-1129/20736*(6*x+5)*(3*x^2+5*x+2)^(1/2)+1129/124416*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3*x^2+5*x
+2)^(1/2))

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maxima [A]  time = 1.31, size = 133, normalized size = 1.00 \begin {gather*} -\frac {4}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {43}{63} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {5017}{1890} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1129}{432} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {5645}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {1129}{3456} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1129}{124416} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {5645}{20736} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/21*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 43/63*(3*x^2 + 5*x + 2)^(5/2)*x + 5017/1890*(3*x^2 + 5*x + 2)^(5/2) + 1129
/432*(3*x^2 + 5*x + 2)^(3/2)*x + 5645/2592*(3*x^2 + 5*x + 2)^(3/2) - 1129/3456*sqrt(3*x^2 + 5*x + 2)*x + 1129/
124416*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 5645/20736*sqrt(3*x^2 + 5*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int {\left (2\,x+3\right )}^2\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 327 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 406 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 185 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 4 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 12 x^{5} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 90 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-327*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-406*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-185*x*
*3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-4*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(12*x**5*sqrt(3*x**2 + 5
*x + 2), x) - Integral(-90*sqrt(3*x**2 + 5*x + 2), x)

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