3.23.64 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx\)

Optimal. Leaf size=150 \[ \frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1080}+\frac {7093 (5 x+3)^{3/2} \sqrt {1-2 x}}{21600}-\frac {390869 \sqrt {5 x+3} \sqrt {1-2 x}}{259200}+\frac {1922677 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {101, 154, 157, 54, 216, 93, 204} \begin {gather*} \frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1080}+\frac {7093 (5 x+3)^{3/2} \sqrt {1-2 x}}{21600}-\frac {390869 \sqrt {5 x+3} \sqrt {1-2 x}}{259200}+\frac {1922677 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x),x]

[Out]

(-390869*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/259200 + (7093*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/21600 + (181*(1 - 2*x)^(3/
2)*(3 + 5*x)^(3/2))/1080 + ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/12 + (1922677*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(
777600*Sqrt[10]) - (98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx &=\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{12} \int \frac {\left (-51-\frac {181 x}{2}\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx\\ &=\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{540} \int \frac {\left (-\frac {5133}{2}-\frac {21279 x}{4}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx\\ &=\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {\int \frac {\left (-\frac {116469}{4}-\frac {1172607 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{16200}\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {\int \frac {\frac {3020277}{8}+\frac {5768031 x}{16}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{97200}\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {1922677 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1555200}+\frac {343}{243} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {686}{243} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1922677 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{777600 \sqrt {5}}\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {1922677 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 118, normalized size = 0.79 \begin {gather*} \frac {30 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (432000 x^3-607200 x^2+230940 x+59599\right )-3136000 \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-1922677 \sqrt {10-20 x} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{7776000 \sqrt {2 x-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x),x]

[Out]

(30*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(59599 + 230940*x - 607200*x^2 + 432000*x^3) - 1922677*Sqrt[10 - 20*x]*Ar
cSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]] - 3136000*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(777
6000*Sqrt[-1 + 2*x])

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IntegrateAlgebraic [A]  time = 0.26, size = 159, normalized size = 1.06 \begin {gather*} -\frac {11 \sqrt {1-2 x} \left (\frac {48858625 (1-2 x)^3}{(5 x+3)^3}-\frac {19807350 (1-2 x)^2}{(5 x+3)^2}-\frac {5785860 (1-2 x)}{5 x+3}-618152\right )}{259200 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {1922677 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x),x]

[Out]

(-11*Sqrt[1 - 2*x]*(-618152 + (48858625*(1 - 2*x)^3)/(3 + 5*x)^3 - (19807350*(1 - 2*x)^2)/(3 + 5*x)^2 - (57858
60*(1 - 2*x))/(3 + 5*x)))/(259200*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^4) - (1922677*ArcTan[(Sqrt[5/2]*
Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(777600*Sqrt[10]) - (98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
243

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fricas [A]  time = 1.46, size = 112, normalized size = 0.75 \begin {gather*} \frac {1}{259200} \, {\left (432000 \, x^{3} - 607200 \, x^{2} + 230940 \, x + 59599\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {49}{243} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {1922677}{15552000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x),x, algorithm="fricas")

[Out]

1/259200*(432000*x^3 - 607200*x^2 + 230940*x + 59599)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 49/243*sqrt(7)*arctan(1/1
4*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 1922677/15552000*sqrt(10)*arctan(1/20*s
qrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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giac [A]  time = 1.52, size = 199, normalized size = 1.33 \begin {gather*} \frac {49}{2430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{1296000} \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 577 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 23769 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 390869 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1922677}{15552000} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x),x, algorithm="giac")

[Out]

49/2430*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2
/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 1/1296000*(12*(8*(36*sqrt(5)*(5*x + 3) - 577*sqrt(5))
*(5*x + 3) + 23769*sqrt(5))*(5*x + 3) - 390869*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 1922677/15552000*sqrt(
10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10
*x + 5) - sqrt(22))))

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maple [A]  time = 0.01, size = 132, normalized size = 0.88 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (25920000 \sqrt {-10 x^{2}-x +3}\, x^{3}-36432000 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x +1922677 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3136000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3575940 \sqrt {-10 x^{2}-x +3}\right )}{15552000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(5*x+3)^(3/2)/(3*x+2),x)

[Out]

1/15552000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(25920000*(-10*x^2-x+3)^(1/2)*x^3-36432000*(-10*x^2-x+3)^(1/2)*x^2+192
2677*10^(1/2)*arcsin(20/11*x+1/11)+3136000*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]
*(-10*x^2-x+3)^(1/2)*x+3575940*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.25, size = 98, normalized size = 0.65 \begin {gather*} -\frac {1}{6} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {271}{1080} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7093}{4320} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1922677}{15552000} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {49}{243} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {135521}{259200} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x),x, algorithm="maxima")

[Out]

-1/6*(-10*x^2 - x + 3)^(3/2)*x + 271/1080*(-10*x^2 - x + 3)^(3/2) + 7093/4320*sqrt(-10*x^2 - x + 3)*x + 192267
7/15552000*sqrt(10)*arcsin(20/11*x + 1/11) + 49/243*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2))
- 135521/259200*sqrt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2),x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(3/2))/(3*x + 2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x),x)

[Out]

Timed out

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