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

Optimal. Leaf size=206 \[ -\frac {1}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac {41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac {3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac {(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac {249299 (6 x+5) \sqrt {3 x^2+5 x+2}}{35831808}-\frac {249299 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{71663616 \sqrt {3}} \]

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Rubi [A]  time = 0.12, antiderivative size = 206, normalized size of antiderivative = 1.00, 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 = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac {41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac {3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac {(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac {249299 (6 x+5) \sqrt {3 x^2+5 x+2}}{35831808}-\frac {249299 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{71663616 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(249299*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/35831808 - (249299*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/4478976 + (2492
99*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/466560 + (3298*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/4455 + (41*(3 + 2*x)
^3*(2 + 5*x + 3*x^2)^(7/2))/110 - ((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2))/33 + ((7405817 + 3365726*x)*(2 + 5*x +
3*x^2)^(7/2))/1496880 - (249299*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(71663616*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)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{33} \int (3+2 x)^3 \left (\frac {1127}{2}+369 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{990} \int (3+2 x)^2 \left (\frac {53829}{2}+19788 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {\int (3+2 x) \left (\frac {1965801}{2}+721227 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx}{26730}\\ &=\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}+\frac {249299 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{12960}\\ &=\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{186624}\\ &=-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}+\frac {249299 \int \sqrt {2+5 x+3 x^2} \, dx}{2985984}\\ &=\frac {249299 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{71663616}\\ &=\frac {249299 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{35831808}\\ &=\frac {249299 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{71663616 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 102, normalized size = 0.50 \begin {gather*} \frac {-95980115 \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 (180592312320 x^{10}[phone] x^9-1932170526720 x^8-25759323039744 x^7-90095929758720 x^6-172473366866688 x^5-204855126595200 x^4-155155370878800 x^3-73069860056520 x^2-19521700361210 x-2261297826735\right )}{82771476480} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-2261297826735 - 19521700361210*x - 73069860056520*x^2 - 155155370878800*x^3 - 2048
55126595200*x^4 - 172473366866688*x^5 - 90095929758720*x^6 - 25759323039744*x^7 - 1932170526720*x^8 + 87587271
4752*x^9 + 180592312320*x^10) - 95980115*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/82771476480

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IntegrateAlgebraic [A]  time = 1.31, size = 104, normalized size = 0.50 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-180592312320 x^{10}-875872714752 x^9+1932170526720 x^8+25759323039744 x^7+90095929758720 x^6+172473366866688 x^5+204855126595200 x^4+155155370878800 x^3+73069860056520 x^2+19521700361210 x+2261297826735\right )}{13795246080}-\frac {249299 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{35831808 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(2261297826735 + 19521700361210*x + 73069860056520*x^2 + 155155370878800*x^3 + 20485512
6595200*x^4 + 172473366866688*x^5 + 90095929758720*x^6 + 25759323039744*x^7 + 1932170526720*x^8 - 875872714752
*x^9 - 180592312320*x^10))/13795246080 - (249299*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(35831808*S
qrt[3])

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fricas [A]  time = 0.41, size = 103, normalized size = 0.50 \begin {gather*} -\frac {1}{13795246080} \, {\left (180592312320 \, x^{10} + 875872714752 \, x^{9} - 1932170526720 \, x^{8} - 25759323039744 \, x^{7} - 90095929758720 \, x^{6} - 172473366866688 \, x^{5} - 204855126595200 \, x^{4} - 155155370878800 \, x^{3} - 73069860056520 \, x^{2} - 19521700361210 \, x - 2261297826735\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {249299}{429981696} \, \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)^4*(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

-1/13795246080*(180592312320*x^10 + 875872714752*x^9 - 1932170526720*x^8 - 25759323039744*x^7 - 90095929758720
*x^6 - 172473366866688*x^5 - 204855126595200*x^4 - 155155370878800*x^3 - 73069860056520*x^2 - 19521700361210*x
- 2261297826735)*sqrt(3*x^2 + 5*x + 2) + 249299/429981696*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 = 0.22, size = 99, normalized size = 0.48 \begin {gather*} -\frac {1}{13795246080} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, {\left (54 \, {\left (20 \, x + 97\right )} x - 11555\right )} x - 7394353\right )} x - 362075335\right )} x - 24952744049\right )} x - 177825630725\right )} x - 1077467853325\right )} x - 3044577502355\right )} x - 9760850180605\right )} x - 2261297826735\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {249299}{214990848} \, \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)^4*(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

-1/13795246080*(2*(12*(6*(8*(6*(36*(14*(48*(54*(20*x + 97)*x - 11555)*x - 7394353)*x - 362075335)*x - 24952744
049)*x - 177825630725)*x - 1077467853325)*x - 3044577502355)*x - 9760850180605)*x - 2261297826735)*sqrt(3*x^2
+ 5*x + 2) + 249299/214990848*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 = 0.06, size = 168, normalized size = 0.82 \begin {gather*} -\frac {16 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{4}}{33}+\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{3}}{55}+\frac {8762 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{2}}{891}+\frac {2642401 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x}{106920}-\frac {249299 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{214990848}+\frac {249299 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{35831808}-\frac {249299 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{4478976}+\frac {249299 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{466560}+\frac {5753773 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{299376} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/33*x^4*(3*x^2+5*x+2)^(7/2)+4/55*x^3*(3*x^2+5*x+2)^(7/2)+8762/891*x^2*(3*x^2+5*x+2)^(7/2)+2642401/106920*x*
(3*x^2+5*x+2)^(7/2)-249299/214990848*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3*x^2+5*x+2)^(1/2))+249299/35831808*(6*
x+5)*(3*x^2+5*x+2)^(1/2)-249299/4478976*(6*x+5)*(3*x^2+5*x+2)^(3/2)+249299/466560*(6*x+5)*(3*x^2+5*x+2)^(5/2)+
5753773/299376*(3*x^2+5*x+2)^(7/2)

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maxima [A]  time = 1.32, size = 196, normalized size = 0.95 \begin {gather*} -\frac {16}{33} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{4} + \frac {4}{55} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {8762}{891} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {2642401}{106920} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {5753773}{299376} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {249299}{77760} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {249299}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {249299}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {1246495}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {249299}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {249299}{214990848} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {1246495}{35831808} \, \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)^4*(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-16/33*(3*x^2 + 5*x + 2)^(7/2)*x^4 + 4/55*(3*x^2 + 5*x + 2)^(7/2)*x^3 + 8762/891*(3*x^2 + 5*x + 2)^(7/2)*x^2 +
2642401/106920*(3*x^2 + 5*x + 2)^(7/2)*x + 5753773/299376*(3*x^2 + 5*x + 2)^(7/2) + 249299/77760*(3*x^2 + 5*x
+ 2)^(5/2)*x + 249299/93312*(3*x^2 + 5*x + 2)^(5/2) - 249299/746496*(3*x^2 + 5*x + 2)^(3/2)*x - 1246495/44789
76*(3*x^2 + 5*x + 2)^(3/2) + 249299/5971968*sqrt(3*x^2 + 5*x + 2)*x - 249299/214990848*sqrt(3)*log(2*sqrt(3)*s
qrt(3*x^2 + 5*x + 2) + 6*x + 5) + 1246495/35831808*sqrt(3*x^2 + 5*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int {\left (2\,x+3\right )}^4\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 12096 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 38421 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 67449 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 70799 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 44295 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 14784 x^{6} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1304 x^{7} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 624 x^{8} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 144 x^{9} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 1620 \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)**4*(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(-12096*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-38421*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-67
449*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-70799*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-44295*x**5*s
qrt(3*x**2 + 5*x + 2), x) - Integral(-14784*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1304*x**7*sqrt(3*x**2
+ 5*x + 2), x) - Integral(624*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(144*x**9*sqrt(3*x**2 + 5*x + 2), x) -
Integral(-1620*sqrt(3*x**2 + 5*x + 2), x)

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