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

Optimal. Leaf size=108 \[ -\frac {17735 \sqrt {1-2 x}}{5929 \sqrt {5 x+3}}-\frac {58}{539 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}+\frac {999 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]

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Rubi [A]  time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {103, 152, 12, 93, 204} \begin {gather*} -\frac {17735 \sqrt {1-2 x}}{5929 \sqrt {5 x+3}}-\frac {58}{539 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}+\frac {999 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-58/(539*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (17735*Sqrt[1 - 2*x])/(5929*Sqrt[3 + 5*x]) + 3/(7*Sqrt[1 - 2*x]*(2 + 3
*x)*Sqrt[3 + 5*x]) + (999*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*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 152

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] && IntegersQ[2*m, 2*n, 2*p]

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}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {1}{7} \int \frac {\frac {31}{2}-60 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {2}{539} \int \frac {-\frac {2503}{4}+435 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {17735 \sqrt {1-2 x}}{5929 \sqrt {3+5 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {4 \int -\frac {120879}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5929}\\ &=-\frac {58}{539 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {17735 \sqrt {1-2 x}}{5929 \sqrt {3+5 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {999}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {17735 \sqrt {1-2 x}}{5929 \sqrt {3+5 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {999}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {58}{539 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {17735 \sqrt {1-2 x}}{5929 \sqrt {3+5 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {999 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 74, normalized size = 0.69 \begin {gather*} \frac {106410 x^2+15821 x-34205}{5929 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}+\frac {999 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-34205 + 15821*x + 106410*x^2)/(5929*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (999*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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IntegrateAlgebraic [A]  time = 0.15, size = 106, normalized size = 0.98 \begin {gather*} \frac {\sqrt {5 x+3} \left (-\frac {12250 (1-2 x)^2}{(5 x+3)^2}-\frac {121671 (1-2 x)}{5 x+3}+112\right )}{5929 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}+7\right )}+\frac {999 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 + 5*x]*(112 - (12250*(1 - 2*x)^2)/(3 + 5*x)^2 - (121671*(1 - 2*x))/(3 + 5*x)))/(5929*Sqrt[1 - 2*x]*(7
+ (1 - 2*x)/(3 + 5*x))) + (999*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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fricas [A]  time = 1.38, size = 101, normalized size = 0.94 \begin {gather*} \frac {120879 \, \sqrt {7} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\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 ) - 14 \, {\left (106410 \, x^{2} + 15821 \, x - 34205\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{83006 \, {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/83006*(120879*sqrt(7)*(30*x^3 + 23*x^2 - 7*x - 6)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 14*(106410*x^2 + 15821*x - 34205)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(30*x^3 + 23*x^2 - 7*x
- 6)

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giac [B]  time = 1.90, size = 278, normalized size = 2.57 \begin {gather*} -\frac {999}{6860} \, \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 {25}{242} \, \sqrt {10} {\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 )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{29645 \, {\left (2 \, x - 1\right )}} - \frac {594 \, \sqrt {10} {\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 )}}{49 \, {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-999/6860*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)))) - 25/242*sqrt(10)*((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))) - 16/29645*sqrt(5)*sqrt(5*x + 3)*sqrt
(-10*x + 5)/(2*x - 1) - 594/49*sqrt(10)*((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)))/(((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)

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maple [B]  time = 0.02, size = 209, normalized size = 1.94 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \left (3626370 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2780217 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1489740 \sqrt {-10 x^{2}-x +3}\, x^{2}-846153 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+221494 \sqrt {-10 x^{2}-x +3}\, x -725274 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-478870 \sqrt {-10 x^{2}-x +3}\right )}{83006 \left (3 x +2\right ) \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/83006*(-2*x+1)^(1/2)*(3626370*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+2780217*7^(1/2
)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-846153*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))+1489740*(-10*x^2-x+3)^(1/2)*x^2-725274*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))+221494*(-10*x^2-x+3)^(1/2)*x-478870*(-10*x^2-x+3)^(1/2))/(3*x+2)/(2*x-1)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [A]  time = 1.30, size = 92, normalized size = 0.85 \begin {gather*} -\frac {999}{686} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {35470 \, x}{5929 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {18373}{5929 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3}{7 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-999/686*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 35470/5929*x/sqrt(-10*x^2 - x + 3) - 1837
3/5929/sqrt(-10*x^2 - x + 3) + 3/7/(3*sqrt(-10*x^2 - x + 3)*x + 2*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 {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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