3.23.6 \(\int (5-x) (3+2 x)^4 (2+5 x+3 x^2)^{7/2} \, dx\)

Optimal. Leaf size=229 \[ -\frac {1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+\frac {439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}}{1404}+\frac {205}{351} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac {(389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}}{227448}+\frac {74167 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{186624}-\frac {519169 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{13436928}+\frac {2595845 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{644972544}-\frac {2595845 (6 x+5) \sqrt {3 x^2+5 x+2}}{5159780352}+\frac {2595845 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{10319560704 \sqrt {3}} \]

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Rubi [A]  time = 0.14, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 10, 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}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+\frac {439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}}{1404}+\frac {205}{351} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac {(389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}}{227448}+\frac {74167 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{186624}-\frac {519169 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{13436928}+\frac {2595845 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{644972544}-\frac {2595845 (6 x+5) \sqrt {3 x^2+5 x+2}}{5159780352}+\frac {2595845 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{10319560704 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(-2595845*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5159780352 + (2595845*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/644972544
- (519169*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/13436928 + (74167*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/186624 + (20
5*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2))/351 + (439*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/1404 - ((3 + 2*x)^4*(2
+ 5*x + 3*x^2)^(9/2))/39 + ((852175 + 389394*x)*(2 + 5*x + 3*x^2)^(9/2))/227448 + (2595845*ArcTanh[(5 + 6*x)/(
2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(10319560704*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 )^{7/2} \, dx &=-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {1}{39} \int (3+2 x)^3 \left (\frac {1337}{2}+439 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {\int (3+2 x)^2 \left (\frac {74595}{2}+27060 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx}{1404}\\ &=\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {\int (3+2 x) \left (\frac {3298845}{2}+1189815 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx}{46332}\\ &=\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac {74167 \int \left (2+5 x+3 x^2\right )^{7/2} \, dx}{3888}\\ &=\frac {74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}-\frac {519169 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{373248}\\ &=-\frac {519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac {74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac {2595845 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{26873856}\\ &=\frac {2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac {519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac {74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}-\frac {2595845 \int \sqrt {2+5 x+3 x^2} \, dx}{429981696}\\ &=-\frac {2595845 (5+6 x) \sqrt {2+5 x+3 x^2}}{5159780352}+\frac {2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac {519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac {74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac {2595845 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{10319560704}\\ &=-\frac {2595845 (5+6 x) \sqrt {2+5 x+3 x^2}}{5159780352}+\frac {2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac {519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac {74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac {2595845 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{5159780352}\\ &=-\frac {2595845 (5+6 x) \sqrt {2+5 x+3 x^2}}{5159780352}+\frac {2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac {519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac {74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac {205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac {1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac {2595845 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{10319560704 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 184, normalized size = 0.80 \begin {gather*} \frac {-36 (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+820 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac {1}{162} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}+\frac {964171 \left (35 \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 (4478976 x^7+26127360 x^6+64800000 x^5+88560000 x^4+72023472 x^3+34858680 x^2+9298342 x+1054785\right )\right )}{286654464}}{1404} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(820*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9/2) + 439*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2) - 36*(3 + 2*x)^4*(2 + 5*x +
3*x^2)^(9/2) + ((852175 + 389394*x)*(2 + 5*x + 3*x^2)^(9/2))/162 + (964171*(6*Sqrt[2 + 5*x + 3*x^2]*(1054785
+ 9298342*x + 34858680*x^2 + 72023472*x^3 + 88560000*x^4 + 64800000*x^5 + 26127360*x^6 + 4478976*x^7) + 35*Sqr
t[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])]))/286654464)/1404

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IntegrateAlgebraic [A]  time = 1.30, size = 114, normalized size = 0.50 \begin {gather*} \frac {2595845 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{5159780352 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-2229025112064 x^{12}-14643456638976 x^{11}[phone] x^{10}[phone] x^9+1590604366381056 x^8+4022427759003648 x^7+6524509131334656 x^6+7203650864723712 x^5+5499074981552256 x^4+2865856228323984 x^3+975104480077800 x^2+195441229635490 x+17510968283403\right )}{67077144576} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 - x)*(3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(17510968283403 + 195441229635490*x + 975104480077800*x^2 + 2865856228323984*x^3 + 5499
074981552256*x^4 + 7203650864723712*x^5 + 6524509131334656*x^6 + 4022427759003648*x^7 + 1590604366381056*x^8 +
333952593887232*x^9 + 2110350163968*x^10 - 14643456638976*x^11 - 2229025112064*x^12))/67077144576 + (2595845*
ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(5159780352*Sqrt[3])

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fricas [A]  time = 0.42, size = 113, normalized size = 0.49 \begin {gather*} -\frac {1}{67077144576} \, {\left (2229025112064 \, x^{12} + 14643456638976 \, x^{11} - 2110350163968 \, x^{10} - 333952593887232 \, x^{9} - 1590604366381056 \, x^{8} - 4022427759003648 \, x^{7} - 6524509131334656 \, x^{6} - 7203650864723712 \, x^{5} - 5499074981552256 \, x^{4} - 2865856228323984 \, x^{3} - 975104480077800 \, x^{2} - 195441229635490 \, x - 17510968283403\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2595845}{61917364224} \, \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)^(7/2),x, algorithm="fricas")

[Out]

-1/67077144576*(2229025112064*x^12 + 14643456638976*x^11 - 2110350163968*x^10 - 333952593887232*x^9 - 15906043
66381056*x^8 - 4022427759003648*x^7 - 6524509131334656*x^6 - 7203650864723712*x^5 - 5499074981552256*x^4 - 286
5856228323984*x^3 - 975104480077800*x^2 - 195441229635490*x - 17510968283403)*sqrt(3*x^2 + 5*x + 2) + 2595845/
61917364224*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 = 0.23, size = 109, normalized size = 0.48 \begin {gather*} -\frac {1}{67077144576} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, {\left (4 \, {\left (6 \, {\left (72 \, x + 473\right )} x - 409\right )} x - 258889\right )} x - 66586273\right )} x - 8082617507\right )} x - 26220538883\right )} x - 1042194858901\right )} x - 4773502588153\right )} x - 19901779363361\right )} x - 40629353336575\right )} x - 97720614817745\right )} x - 17510968283403\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {2595845}{30958682112} \, \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)^(7/2),x, algorithm="giac")

[Out]

-1/67077144576*(2*(12*(6*(8*(6*(36*(2*(48*(54*(4*(6*(72*x + 473)*x - 409)*x - 258889)*x - 66586273)*x - 808261
7507)*x - 26220538883)*x - 1042194858901)*x - 4773502588153)*x - 19901779363361)*x - 40629353336575)*x - 97720
614817745)*x - 17510968283403)*sqrt(3*x^2 + 5*x + 2) - 2595845/30958682112*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 = 187, normalized size = 0.82 \begin {gather*} -\frac {16 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x^{4}}{39}+\frac {14 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x^{3}}{351}+\frac {2827 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x^{2}}{351}+\frac {84521 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x}{4212}+\frac {2595845 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{30958682112}+\frac {74167 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{186624}-\frac {2595845 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{5159780352}+\frac {2595845 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{644972544}-\frac {519169 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{13436928}+\frac {3495529 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{227448} \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)^(7/2),x)

[Out]

-16/39*x^4*(3*x^2+5*x+2)^(9/2)+14/351*x^3*(3*x^2+5*x+2)^(9/2)+2827/351*x^2*(3*x^2+5*x+2)^(9/2)+84521/4212*x*(3
*x^2+5*x+2)^(9/2)+74167/186624*(6*x+5)*(3*x^2+5*x+2)^(7/2)+2595845/30958682112*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2
)+(3*x^2+5*x+2)^(1/2))-2595845/5159780352*(6*x+5)*(3*x^2+5*x+2)^(1/2)+2595845/644972544*(6*x+5)*(3*x^2+5*x+2)^
(3/2)-519169/13436928*(6*x+5)*(3*x^2+5*x+2)^(5/2)+3495529/227448*(3*x^2+5*x+2)^(9/2)

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maxima [A]  time = 1.34, size = 225, normalized size = 0.98 \begin {gather*} -\frac {16}{39} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{4} + \frac {14}{351} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{3} + \frac {2827}{351} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{2} + \frac {84521}{4212} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {3495529}{227448} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {74167}{31104} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {370835}{186624} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {519169}{2239488} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {2595845}{13436928} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {2595845}{107495424} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {12979225}{644972544} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {2595845}{859963392} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {2595845}{30958682112} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {12979225}{5159780352} \, \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)^(7/2),x, algorithm="maxima")

[Out]

-16/39*(3*x^2 + 5*x + 2)^(9/2)*x^4 + 14/351*(3*x^2 + 5*x + 2)^(9/2)*x^3 + 2827/351*(3*x^2 + 5*x + 2)^(9/2)*x^2
+ 84521/4212*(3*x^2 + 5*x + 2)^(9/2)*x + 3495529/227448*(3*x^2 + 5*x + 2)^(9/2) + 74167/31104*(3*x^2 + 5*x +
2)^(7/2)*x + 370835/186624*(3*x^2 + 5*x + 2)^(7/2) - 519169/2239488*(3*x^2 + 5*x + 2)^(5/2)*x - 2595845/134369
28*(3*x^2 + 5*x + 2)^(5/2) + 2595845/107495424*(3*x^2 + 5*x + 2)^(3/2)*x + 12979225/644972544*(3*x^2 + 5*x + 2
)^(3/2) - 2595845/859963392*sqrt(3*x^2 + 5*x + 2)*x + 2595845/30958682112*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5
*x + 2) + 6*x + 5) - 12979225/5159780352*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 )}^{7/2} \,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)^(7/2),x)

[Out]

-int((2*x + 3)^4*(x - 5)*(5*x + 3*x^2 + 2)^(7/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 32292 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 142182 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 363291 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 594106 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 644932 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 463440 x^{6} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 209413 x^{7} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 49624 x^{8} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 504 x^{9} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 2592 x^{10} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 432 x^{11} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 3240 \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)**(7/2),x)

[Out]

-Integral(-32292*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-142182*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3
63291*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-594106*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-644932*x*
*5*sqrt(3*x**2 + 5*x + 2), x) - Integral(-463440*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-209413*x**7*sqrt(
3*x**2 + 5*x + 2), x) - Integral(-49624*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral(-504*x**9*sqrt(3*x**2 + 5*x
+ 2), x) - Integral(2592*x**10*sqrt(3*x**2 + 5*x + 2), x) - Integral(432*x**11*sqrt(3*x**2 + 5*x + 2), x) - I
ntegral(-3240*sqrt(3*x**2 + 5*x + 2), x)

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