3.2578 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=130 \[ \frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {5 x+3}}-\frac {28705 \sqrt {1-2 x}}{17787 (5 x+3)^{3/2}}-\frac {58}{539 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac {4887 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]

[Out]

-4887/343*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-58/539/(3+5*x)^(3/2)/(1-2*x)^(1/2)+3/7/(2+3*
x)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-28705/17787*(1-2*x)^(1/2)/(3+5*x)^(3/2)+2841815/195657*(1-2*x)^(1/2)/(3+5*x)^(1
/2)

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Rubi [A]  time = 0.05, antiderivative size = 130, 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 = {103, 152, 12, 93, 204} \[ \frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {5 x+3}}-\frac {28705 \sqrt {1-2 x}}{17787 (5 x+3)^{3/2}}-\frac {58}{539 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac {4887 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-58/(539*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (28705*Sqrt[1 - 2*x])/(17787*(3 + 5*x)^(3/2)) + 3/(7*Sqrt[1 - 2*x]*(
2 + 3*x)*(3 + 5*x)^(3/2)) + (2841815*Sqrt[1 - 2*x])/(195657*Sqrt[3 + 5*x]) - (4887*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)^{5/2}} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {1}{7} \int \frac {\frac {61}{2}-90 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}-\frac {2}{539} \int \frac {-\frac {3653}{4}+870 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {4 \int \frac {-\frac {361687}{8}+\frac {86115 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{17787}\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {8 \int -\frac {19513791}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{195657}\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}+\frac {4887}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}+\frac {4887}{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} (3+5 x)^{3/2}}-\frac {28705 \sqrt {1-2 x}}{17787 (3+5 x)^{3/2}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac {2841815 \sqrt {1-2 x}}{195657 \sqrt {3+5 x}}-\frac {4887 \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 = 102, normalized size = 0.78 \[ \frac {-19513791 \sqrt {7-14 x} \sqrt {5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-7 \left (85254450 x^3+63467215 x^2-20145298 x-16461125\right )}{1369599 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(-16461125 - 20145298*x + 63467215*x^2 + 85254450*x^3) - 19513791*Sqrt[7 - 14*x]*Sqrt[3 + 5*x]*(6 + 19*x +
15*x^2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1369599*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2))

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fricas [A]  time = 1.40, size = 116, normalized size = 0.89 \[ -\frac {19513791 \, \sqrt {7} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (85254450 \, x^{3} + 63467215 \, x^{2} - 20145298 \, x - 16461125\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2739198 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

-1/2739198*(19513791*sqrt(7)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(85254450*x^3 + 63467215*x^2 - 20145298*x - 16461125)*sqrt(5*x + 3
)*sqrt(-2*x + 1))/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)

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giac [B]  time = 2.22, size = 335, normalized size = 2.58 \[ \frac {4887}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{63888} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {1488 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {5952 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{326095 \, {\left (2 \, x - 1\right )}} + \frac {1782 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]

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)^(5/2),x, algorithm="giac")

[Out]

4887/6860*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)))) - 25/63888*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 - 1488*(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))/sqrt(5*x + 3) + 5952*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 32/326095*sqrt(5)*sqrt(
5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1782/49*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*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)

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maple [B]  time = 0.02, size = 257, normalized size = 1.98 \[ \frac {\sqrt {-2 x +1}\, \left (2927068650 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}-995203341 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-282034172 \sqrt {-10 x^{2}-x +3}\, x -351248238 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-230455750 \sqrt {-10 x^{2}-x +3}\right )}{2739198 \left (3 x +2\right ) \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2739198*(-2*x+1)^(1/2)*(2927068650*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))[phone]*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^3-995203341*7^(1/2)*x*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^2-351248238*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))-282034172*(-10*x^2-x+3)^(1/2)*x-230455750*(-10*x^2-x+3)^(1/2))/(3*x+2)/(2*x-1)/(-10*x^2-x+3)^(
1/2)/(5*x+3)^(3/2)

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

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)^(5/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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