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

Optimal. Leaf size=93 \[ \frac {333 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}-\frac {3827 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \]

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Rubi [A]  time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {103, 151, 12, 93, 204} \begin {gather*} \frac {333 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}-\frac {3827 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (333*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)) - (3827*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

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])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx &=\frac {3 \sqrt {1-2 x} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {1}{14} \int \frac {\frac {71}{2}-30 x}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {1-2 x} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {333 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {1}{98} \int \frac {3827}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {1-2 x} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {333 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3827}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {1-2 x} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {333 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3827}{196} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {3 \sqrt {1-2 x} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {333 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}-\frac {3827 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 69, normalized size = 0.74 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (333 x+236)}{(3 x+2)^2}-3827 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(236 + 333*x))/(2 + 3*x)^2 - 3827*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[
3 + 5*x])])/1372

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IntegrateAlgebraic [A]  time = 0.18, size = 90, normalized size = 0.97 \begin {gather*} \frac {33 \sqrt {1-2 x} \left (\frac {181 (1-2 x)}{5 x+3}+805\right )}{196 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^2}-\frac {3827 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(33*Sqrt[1 - 2*x]*(805 + (181*(1 - 2*x))/(3 + 5*x)))/(196*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^2) - (3827*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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fricas [A]  time = 1.36, size = 86, normalized size = 0.92 \begin {gather*} -\frac {3827 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \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 ) - 42 \, {\left (333 \, x + 236\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2744 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2744*(3827*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2
+ x - 3)) - 42*(333*x + 236)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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giac [B]  time = 1.45, size = 252, normalized size = 2.71 \begin {gather*} \frac {3827}{27440} \, \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 {33 \, \sqrt {10} {\left (181 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {32200 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {128800 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3827/27440*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)))) + 33/98*sqrt(10)*(181*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 32200*(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) - 128800*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2

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maple [B]  time = 0.02, size = 154, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (34443 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+45924 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13986 \sqrt {-10 x^{2}-x +3}\, x +15308 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+9912 \sqrt {-10 x^{2}-x +3}\right )}{2744 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2744*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(34443*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4592
4*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+15308*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))+13986*(-10*x^2-x+3)^(1/2)*x+9912*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^2

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maxima [A]  time = 1.11, size = 76, normalized size = 0.82 \begin {gather*} \frac {3827}{2744} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3 \, \sqrt {-10 \, x^{2} - x + 3}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {333 \, \sqrt {-10 \, x^{2} - x + 3}}{196 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3827/2744*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3/14*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x
+ 4) + 333/196*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [B]  time = 13.23, size = 1037, normalized size = 11.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((11841*((1 - 2*x)^(1/2) - 1)^5)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^5) - (23682*((1 - 2*x)^(1/2) - 1)^3)/(6125*
(3^(1/2) - (5*x + 3)^(1/2))^3) - (7458*((1 - 2*x)^(1/2) - 1))/(30625*(3^(1/2) - (5*x + 3)^(1/2))) + (3729*((1
- 2*x)^(1/2) - 1)^7)/(980*(3^(1/2) - (5*x + 3)^(1/2))^7) + (34149*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(30625*(3^(
1/2) - (5*x + 3)^(1/2))^2) - (58782*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(30625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (
34149*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(4900*(3^(1/2) - (5*x + 3)^(1/2))^6))/((544*((1 - 2*x)^(1/2) - 1)^2)/(6
25*(3^(1/2) - (5*x + 3)^(1/2))^2) - (1764*((1 - 2*x)^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (136*
((1 - 2*x)^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^(1/2) - (5*x + 3)^(1/
2))^8 - (96*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(625*(3^(1/2) - (5*x + 3)^(1/2))^3) + (48*3^(1/2)*((1 - 2*x)^(1/2
) - 1)^5)/(125*(3^(1/2) - (5*x + 3)^(1/2))^5) + (12*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(5*(3^(1/2) - (5*x + 3)^(
1/2))^7) - (96*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(625*(3^(1/2) - (5*x + 3)^(1/2))) + 16/625) - (3827*7^(1/2)*atan
(((3827*7^(1/2)*((22962*3^(1/2))/6125 + (11481*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^(1/2))) - (7^
(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/
(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*3827i)/2744 - (11481*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1
/2) - (5*x + 3)^(1/2))^2)))/2744 + (3827*7^(1/2)*((22962*3^(1/2))/6125 + (11481*((1 - 2*x)^(1/2) - 1))/(6125*(
3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (88
8*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*3827i)/2744 - (11481*3^(1/2)*((1
- 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2)))/2744)/((7^(1/2)*((22962*3^(1/2))/6125 + (11481*((
1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2)
- (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*3827
i)/2744 - (11481*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2))*3827i)/2744 - (7^(1/2)
*((22962*3^(1/2))/6125 + (11481*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1
- 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) -
(5*x + 3)^(1/2))) - 536/125)*3827i)/2744 - (11481*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)
^(1/2))^2))*3827i)/2744 + (14645929*((1 - 2*x)^(1/2) - 1)^2)/(480200*(3^(1/2) - (5*x + 3)^(1/2))^2) + 14645929
/1200500)))/1372

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{3} \sqrt {5 x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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