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

Optimal. Leaf size=169 \[ \frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}-\frac {(7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}}{6912}+\frac {5 (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}}{331776}+\frac {5 (1229315-2568342 x) \sqrt {3 x^2+5 x+2}}{2654208}-\frac {65251715 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{5308416 \sqrt {3}}+\frac {1625}{512} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 169, 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 = {814, 843, 621, 206, 724} \begin {gather*} \frac {1}{672} (277-42 x) \left (3 x^2+5 x+2\right )^{7/2}-\frac {(7446 x+589) \left (3 x^2+5 x+2\right )^{5/2}}{6912}+\frac {5 (6205-127338 x) \left (3 x^2+5 x+2\right )^{3/2}}{331776}+\frac {5 (1229315-2568342 x) \sqrt {3 x^2+5 x+2}}{2654208}-\frac {65251715 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{5308416 \sqrt {3}}+\frac {1625}{512} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(1229315 - 2568342*x)*Sqrt[2 + 5*x + 3*x^2])/2654208 + (5*(6205 - 127338*x)*(2 + 5*x + 3*x^2)^(3/2))/331776
- ((589 + 7446*x)*(2 + 5*x + 3*x^2)^(5/2))/6912 + ((277 - 42*x)*(2 + 5*x + 3*x^2)^(7/2))/672 - (65251715*ArcT
anh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(5308416*Sqrt[3]) + (1625*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[
5]*Sqrt[2 + 5*x + 3*x^2])])/512

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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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} \, dx &=\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{192} \int \frac {(2163+2482 x) \left (2+5 x+3 x^2\right )^{5/2}}{3+2 x} \, dx\\ &=-\frac {(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac {\int \frac {(-355890-424460 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx}{27648}\\ &=\frac {5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac {(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {\int \frac {(43354260+51366840 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx}{2654208}\\ &=\frac {5 (1229315-2568342 x) \sqrt {2+5 x+3 x^2}}{2654208}+\frac {5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac {(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}+\frac {\int \frac {-2676363480-3132082320 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{127401984}\\ &=\frac {5 (1229315-2568342 x) \sqrt {2+5 x+3 x^2}}{2654208}+\frac {5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac {(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {65251715 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{5308416}+\frac {8125}{512} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {5 (1229315-2568342 x) \sqrt {2+5 x+3 x^2}}{2654208}+\frac {5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac {(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {65251715 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{2654208}-\frac {8125}{256} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {5 (1229315-2568342 x) \sqrt {2+5 x+3 x^2}}{2654208}+\frac {5 (6205-127338 x) \left (2+5 x+3 x^2\right )^{3/2}}{331776}-\frac {(589+7446 x) \left (2+5 x+3 x^2\right )^{5/2}}{6912}+\frac {1}{672} (277-42 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {65251715 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{5308416 \sqrt {3}}+\frac {1625}{512} \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 123, normalized size = 0.73 \begin {gather*} \frac {-353808000 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-456762005 \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 (31352832 x^7-50015232 x^6-529784064 x^5-1167854976 x^4-1224844848 x^3-722869752 x^2-185981750 x-101435865\right )}{111476736} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-101435865 - 185981750*x - 722869752*x^2 - 1224844848*x^3 - 1167854976*x^4 - 529784
064*x^5 - 50015232*x^6 + 31352832*x^7) - 353808000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)] - 456762005*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/111476736

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IntegrateAlgebraic [A]  time = 0.98, size = 124, normalized size = 0.73 \begin {gather*} -\frac {65251715 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{2654208 \sqrt {3}}+\frac {1625}{256} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {\sqrt {3 x^2+5 x+2} \left (-31352832 x^7+50015232 x^6+529784064 x^5+1167854976 x^4+1224844848 x^3+722869752 x^2+185981750 x+101435865\right )}{18579456} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(101435865 + 185981750*x + 722869752*x^2 + 1224844848*x^3 + 1167854976*x^4 + 529784064*
x^5 + 50015232*x^6 - 31352832*x^7))/18579456 - (65251715*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(26
54208*Sqrt[3]) + (1625*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/256

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fricas [A]  time = 0.42, size = 139, normalized size = 0.82 \begin {gather*} -\frac {1}{18579456} \, {\left (31352832 \, x^{7} - 50015232 \, x^{6} - 529784064 \, x^{5} - 1167854976 \, x^{4} - 1224844848 \, x^{3} - 722869752 \, x^{2} - 185981750 \, x - 101435865\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {65251715}{31850496} \, \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 ) + \frac {1625}{1024} \, \sqrt {5} \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 ) \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),x, algorithm="fricas")

[Out]

-1/18579456*(31352832*x^7 - 50015232*x^6 - 529784064*x^5 - 1167854976*x^4 - 1224844848*x^3 - 722869752*x^2 - 1
85981750*x - 101435865)*sqrt(3*x^2 + 5*x + 2) + 65251715/31850496*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)
*(6*x + 5) + 72*x^2 + 120*x + 49) + 1625/1024*sqrt(5)*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))

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giac [A]  time = 0.36, size = 156, normalized size = 0.92 \begin {gather*} -\frac {1}{18579456} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (42 \, x - 67\right )} x - 25549\right )} x - 337921\right )} x - 2835289\right )} x - 30119573\right )} x - 92990875\right )} x - 101435865\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1625}{512} \, \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 {65251715}{15925248} \, \sqrt {3} \log \left ({\left | -6 \, \sqrt {3} x - 5 \, \sqrt {3} + 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\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),x, algorithm="giac")

[Out]

-1/18579456*(2*(12*(18*(8*(6*(36*(42*x - 67)*x - 25549)*x - 337921)*x - 2835289)*x - 30119573)*x - 92990875)*x
- 101435865)*sqrt(3*x^2 + 5*x + 2) + 1625/512*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))) + 65251715/15925248*sqrt
(3)*log(abs(-6*sqrt(3)*x - 5*sqrt(3) + 6*sqrt(3*x^2 + 5*x + 2)))

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maple [B]  time = 0.05, size = 295, normalized size = 1.75 \begin {gather*} -\frac {1625 \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 )}{512}-\frac {679705 \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 )}{165888}-\frac {35 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{15925248}-\frac {\left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{96}+\frac {7 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{6912}-\frac {35 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{331776}+\frac {35 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{2654208}+\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{28}-\frac {13 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{72}-\frac {1105 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{3456}-\frac {22295 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{27648}+\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{16}+\frac {325 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{192}+\frac {1625 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{512} \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),x)

[Out]

-1/96*(6*x+5)*(3*x^2+5*x+2)^(7/2)+7/6912*(6*x+5)*(3*x^2+5*x+2)^(5/2)-35/331776*(6*x+5)*(3*x^2+5*x+2)^(3/2)+35/
2654208*(6*x+5)*(3*x^2+5*x+2)^(1/2)-35/15925248*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3*x^2+5*x+2)^(1/2))+13/28*(-
4*x+3*(x+3/2)^2-19/4)^(7/2)-13/72*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1105/3456*(6*x+5)*(-4*x+3*(x+3/2)^2-19
/4)^(3/2)-22295/27648*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-679705/165888*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4
*x+3*(x+3/2)^2-19/4)^(1/2))+13/16*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+325/192*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+1625/512
*(-16*x+12*(x+3/2)^2-19)^(1/2)-1625/512*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))

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maxima [A]  time = 1.31, size = 186, normalized size = 1.10 \begin {gather*} -\frac {1}{16} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {277}{672} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {1241}{1152} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {589}{6912} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {106115}{55296} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {31025}{331776} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {2140285}{442368} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {65251715}{15925248} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {1625}{512} \, \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 {6146575}{2654208} \, \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),x, algorithm="maxima")

[Out]

-1/16*(3*x^2 + 5*x + 2)^(7/2)*x + 277/672*(3*x^2 + 5*x + 2)^(7/2) - 1241/1152*(3*x^2 + 5*x + 2)^(5/2)*x - 589/
6912*(3*x^2 + 5*x + 2)^(5/2) - 106115/55296*(3*x^2 + 5*x + 2)^(3/2)*x + 31025/331776*(3*x^2 + 5*x + 2)^(3/2) -
2140285/442368*sqrt(3*x^2 + 5*x + 2)*x - 65251715/15925248*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x +
5/2) - 1625/512*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 6146575/26542
08*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}}{2\,x+3} \,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),x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3), 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}}{2 x + 3}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\, 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),x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - In
tegral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x + 3),
x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(2*x
+ 3), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x)

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