3.23.16 \(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=197 \[ -\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac {7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac {7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac {21 (21974 x+47145) \sqrt {3 x^2+5 x+2}}{10240 (2 x+3)}+\frac {30275 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024}-\frac {2345091 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{20480 \sqrt {5}} \]

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Rubi [A]  time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {812, 810, 843, 621, 206, 724} \begin {gather*} -\frac {(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac {7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac {7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac {21 (21974 x+47145) \sqrt {3 x^2+5 x+2}}{10240 (2 x+3)}+\frac {30275 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024}-\frac {2345091 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{20480 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6,x]

[Out]

(-21*(47145 + 21974*x)*Sqrt[2 + 5*x + 3*x^2])/(10240*(3 + 2*x)) + (7*(42733 + 33142*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(7680*(3 + 2*x)^3) + (7*(1003 + 548*x)*(2 + 5*x + 3*x^2)^(5/2))/(960*(3 + 2*x)^4) - ((27 + 5*x)*(2 + 5*x + 3
*x^2)^(7/2))/(30*(3 + 2*x)^5) + (30275*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/1024 - (2
345091*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(20480*Sqrt[5])

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 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 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]

Rule 810

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

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 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]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx &=-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}-\frac {7}{120} \int \frac {(-230-274 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx\\ &=\frac {7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac {7 \int \frac {(-11292-13112 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{1536}\\ &=\frac {7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac {7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}-\frac {7 \int \frac {(1351944+1582128 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{122880}\\ &=-\frac {21 (47145+21974 x) \sqrt {2+5 x+3 x^2}}{10240 (3+2 x)}+\frac {7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac {7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac {7 \int \frac {21287376+24912000 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{983040}\\ &=-\frac {21 (47145+21974 x) \sqrt {2+5 x+3 x^2}}{10240 (3+2 x)}+\frac {7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac {7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac {90825 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1024}-\frac {2345091 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{20480}\\ &=-\frac {21 (47145+21974 x) \sqrt {2+5 x+3 x^2}}{10240 (3+2 x)}+\frac {7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac {7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac {90825}{512} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {2345091 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{10240}\\ &=-\frac {21 (47145+21974 x) \sqrt {2+5 x+3 x^2}}{10240 (3+2 x)}+\frac {7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac {7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac {(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac {30275 \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1024}-\frac {2345091 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{20480 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 130, normalized size = 0.66 \begin {gather*} \frac {2345091 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+3027500 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {10 \sqrt {3 x^2+5 x+2} \left (46080 x^7-257280 x^6+483840 x^5+27897856 x^4+127665096 x^3+242016116 x^2+213122626 x+72189541\right )}{(2 x+3)^5}}{102400} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6,x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(72189541 + 213122626*x + 242016116*x^2 + 127665096*x^3 + 27897856*x^4 + 483840*x^
5 - 257280*x^6 + 46080*x^7))/(3 + 2*x)^5 + 2345091*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)] + 3027500*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/102400

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IntegrateAlgebraic [A]  time = 1.03, size = 131, normalized size = 0.66 \begin {gather*} \frac {30275}{512} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )-\frac {2345091 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{10240 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-46080 x^7+257280 x^6-483840 x^5-27897856 x^4-127665096 x^3-242016116 x^2-213122626 x-72189541\right )}{10240 (2 x+3)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-72189541 - 213122626*x - 242016116*x^2 - 127665096*x^3 - 27897856*x^4 - 483840*x^5 +
257280*x^6 - 46080*x^7))/(10240*(3 + 2*x)^5) + (30275*Sqrt[3]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))]
)/512 - (2345091*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(10240*Sqrt[5])

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fricas [A]  time = 0.42, size = 219, normalized size = 1.11 \begin {gather*} \frac {3027500 \, \sqrt {3} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 2345091 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \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 ) - 20 \, {\left (46080 \, x^{7} - 257280 \, x^{6} + 483840 \, x^{5} + 27897856 \, x^{4} + 127665096 \, x^{3} + 242016116 \, x^{2} + 213122626 \, x + 72189541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{204800 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}

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)^6,x, algorithm="fricas")

[Out]

1/204800*(3027500*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x
+ 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 2345091*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
*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)) - 20*(46080*x^7 -
257280*x^6 + 483840*x^5 + 27897856*x^4 + 127665096*x^3 + 242016116*x^2 + 213122626*x + 72189541)*sqrt(3*x^2 +
5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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giac [B]  time = 0.37, size = 417, normalized size = 2.12 \begin {gather*} -\frac {3}{512} \, {\left (2 \, {\left (12 \, x - 157\right )} x + 2067\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {2345091}{102400} \, \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 {30275}{1024} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {60397264 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 739203704 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11836231432 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 36096211012 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 207702455456 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 259725515674 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 635418284542 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 326158305587 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 287216072451 \, \sqrt {3} x + 36785380096 \, \sqrt {3} - 287216072451 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{10240 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \end {gather*}

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)^6,x, algorithm="giac")

[Out]

-3/512*(2*(12*x - 157)*x + 2067)*sqrt(3*x^2 + 5*x + 2) - 2345091/102400*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)))
- 30275/1024*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 1/10240*(60397264*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2))^9 + 739203704*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 11836231432*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2))^7 + 36096211012*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 207702455456*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 259725515674*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 635418284542*
(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 326158305587*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 287216072
451*sqrt(3)*x + 36785380096*sqrt(3) - 287216072451*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2
))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^5

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maple [A]  time = 0.06, size = 316, normalized size = 1.60 \begin {gather*} \frac {2345091 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{102400}+\frac {30275 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{1024}-\frac {27 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{8000 \left (x +\frac {3}{2}\right )^{4}}+\frac {10023 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{100000 \left (x +\frac {3}{2}\right )^{2}}-\frac {251 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{5000 \left (x +\frac {3}{2}\right )^{3}}+\frac {19059 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{25000}+\frac {122871 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{100000}-\frac {19059 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{12500 \left (x +\frac {3}{2}\right )}+\frac {37037 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{16000}+\frac {37233 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{6400}-\frac {2345091 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{102400}-\frac {781697 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{64000}-\frac {2345091 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{400000}-\frac {335013 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{100000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(2*x+3)^6,x)

[Out]

-27/8000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+10023/100000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-251/5000
/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+19059/25000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+122871/100000*(6*x+
5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-19059/12500/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+37037/16000*(6*x+5)*(-4*x+3
*(x+3/2)^2-19/4)^(3/2)+37233/6400*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)+2345091/102400*5^(1/2)*arctanh(2/5*(-4
*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+30275/1024*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-1
9/4)^(1/2))-2345091/102400*(-16*x+12*(x+3/2)^2-19)^(1/2)-781697/64000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-2345091/40
0000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-335013/100000*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-13/800/(x+3/2)^5*(-4*x+3*(x+3/2
)^2-19/4)^(9/2)

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maxima [B]  time = 1.29, size = 326, normalized size = 1.65 \begin {gather*} -\frac {30069}{100000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {251 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{625 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac {10023 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{25000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {368613}{50000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {112329}{400000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {19059 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5000 \, {\left (2 \, x + 3\right )}} + \frac {111111}{8000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {40957}{64000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {111699}{3200} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {30275}{1024} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {2345091}{102400} \, \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 {855771}{51200} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

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)^6,x, algorithm="maxima")

[Out]

-30069/100000*(3*x^2 + 5*x + 2)^(7/2) - 13/25*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 +
810*x + 243) - 27/500*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 251/625*(3*x^2 + 5*x
+ 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 10023/25000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 368613/500
00*(3*x^2 + 5*x + 2)^(5/2)*x + 112329/400000*(3*x^2 + 5*x + 2)^(5/2) - 19059/5000*(3*x^2 + 5*x + 2)^(7/2)/(2*x
+ 3) + 111111/8000*(3*x^2 + 5*x + 2)^(3/2)*x - 40957/64000*(3*x^2 + 5*x + 2)^(3/2) + 111699/3200*sqrt(3*x^2 +
5*x + 2)*x + 30275/1024*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 2345091/102400*sqrt(5)*log(s
qrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 855771/51200*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 \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^6,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \end {gather*}

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)**6,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729),
x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x +
729), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2
+ 2916*x + 729), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 +
4860*x**2 + 2916*x + 729), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 +
4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 21
60*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x
**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x)

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