3.2634 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac {1001590 \sqrt {1-2 x}}{2152227 \sqrt {5 x+3}}-\frac {19130 \sqrt {1-2 x}}{195657 (5 x+3)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {162 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]

[Out]

4/231/(1-2*x)^(3/2)/(3+5*x)^(3/2)-162/343*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+412/5929/(3+
5*x)^(3/2)/(1-2*x)^(1/2)-19130/195657*(1-2*x)^(1/2)/(3+5*x)^(3/2)+1001590/2152227*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {104, 152, 12, 93, 204} \[ \frac {1001590 \sqrt {1-2 x}}{2152227 \sqrt {5 x+3}}-\frac {19130 \sqrt {1-2 x}}{195657 (5 x+3)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {162 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 412/(5929*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (19130*Sqrt[1 - 2*x])/(19
5657*(3 + 5*x)^(3/2)) + (1001590*Sqrt[1 - 2*x])/(2152227*Sqrt[3 + 5*x]) - (162*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[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 104

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] &&
IntegersQ[2*m, 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)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {219}{2}-90 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {4 \int \frac {\frac {27987}{4}+9270 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {19130 \sqrt {1-2 x}}{195657 (3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {93873}{8}-\frac {86085 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{586971}\\ &=\frac {4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {19130 \sqrt {1-2 x}}{195657 (3+5 x)^{3/2}}+\frac {1001590 \sqrt {1-2 x}}{2152227 \sqrt {3+5 x}}+\frac {16 \int \frac {10673289}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6456681}\\ &=\frac {4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {19130 \sqrt {1-2 x}}{195657 (3+5 x)^{3/2}}+\frac {1001590 \sqrt {1-2 x}}{2152227 \sqrt {3+5 x}}+\frac {81}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {19130 \sqrt {1-2 x}}{195657 (3+5 x)^{3/2}}+\frac {1001590 \sqrt {1-2 x}}{2152227 \sqrt {3+5 x}}+\frac {162}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {412}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {19130 \sqrt {1-2 x}}{195657 (3+5 x)^{3/2}}+\frac {1001590 \sqrt {1-2 x}}{2152227 \sqrt {3+5 x}}-\frac {162 \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 = 0.13, size = 93, normalized size = 0.76 \[ \frac {7115526 \sqrt {7-14 x} \sqrt {5 x+3} \left (10 x^2+x-3\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+14 \left (10015900 x^3-4427220 x^2-3234261 x+1490582\right )}{15065589 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1490582 - 3234261*x - 4427220*x^2 + 10015900*x^3) + 7115526*Sqrt[7 - 14*x]*Sqrt[3 + 5*x]*(-3 + x + 10*x^2
)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(15065589*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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fricas [A]  time = 1.06, size = 116, normalized size = 0.94 \[ -\frac {3557763 \, \sqrt {7} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (10015900 \, x^{3} - 4427220 \, x^{2} - 3234261 \, x + 1490582\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{15065589 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15065589*(3557763*sqrt(7)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(10015900*x^3 - 4427220*x^2 - 3234261*x + 1490582)*sqrt(5*x + 3)*sqrt
(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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giac [B]  time = 1.48, size = 229, normalized size = 1.86 \[ \frac {81}{3430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{702768} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {648 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2592 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {32 \, {\left (379 \, \sqrt {5} {\left (5 \, x + 3\right )} - 2277 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{53805675 \, {\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/3430*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)))) - 25/702768*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)))^3 - 648*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3) + 2592*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 32/53805675*(379*sqrt(5)*
(5*x + 3) - 2277*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [B]  time = 0.02, size = 250, normalized size = 2.03 \[ \frac {\sqrt {-2 x +1}\, \left (355776300 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}-209908017 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-61981080 \sqrt {-10 x^{2}-x +3}\, x^{2}-21346578 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-45279654 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{15065589 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15065589*(-2*x+1)^(1/2)*(355776300*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7
^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-209908017*7^(1/2)*x^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^3-21346578*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))-61981080*(-10*x^2-x+3)^(1/2)*x^2+32019867*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))-45279654*(-10*x^2-x+3)^(1/2)*x+20868148*(-10*x^2-x+3)^(1/2))/(2*x-1)^2/(-10*x^2-x+3)^(1/2)/(5*x+3)^(
3/2)

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maxima [A]  time = 1.52, size = 87, normalized size = 0.71 \[ \frac {81}{343} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2003180 \, x}{2152227 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1085762}{2152227 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {740 \, x}{2541 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {326}{2541 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/343*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2003180/2152227*x/sqrt(-10*x^2 - x + 3) + 1
085762/2152227/sqrt(-10*x^2 - x + 3) + 740/2541*x/(-10*x^2 - x + 3)^(3/2) - 326/2541/(-10*x^2 - x + 3)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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