3.22.78 \(\int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {5 x+3}}+\frac {14101 \sqrt {1-2 x}}{392 (3 x+2) \sqrt {5 x+3}}+\frac {81 \sqrt {1-2 x}}{28 (3 x+2)^2 \sqrt {5 x+3}}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 \sqrt {5 x+3}}+\frac {1463447 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \]

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Rubi [A]  time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 151, 152, 12, 93, 204} \begin {gather*} -\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {5 x+3}}+\frac {14101 \sqrt {1-2 x}}{392 (3 x+2) \sqrt {5 x+3}}+\frac {81 \sqrt {1-2 x}}{28 (3 x+2)^2 \sqrt {5 x+3}}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 \sqrt {5 x+3}}+\frac {1463447 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

(-639565*Sqrt[1 - 2*x])/(1176*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (81*Sqrt[1 - 2*x]
)/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (14101*Sqrt[1 - 2*x])/(392*(2 + 3*x)*Sqrt[3 + 5*x]) + (1463447*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*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 99

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

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 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 {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}-\frac {1}{3} \int \frac {-\frac {41}{2}+30 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}-\frac {1}{42} \int \frac {-\frac {7621}{4}+2430 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}-\frac {1}{294} \int \frac {-\frac {899407}{8}+\frac {211515 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}+\frac {\int -\frac {48293751}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1617}\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}-\frac {1463447}{784} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}-\frac {1463447}{392} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {639565 \sqrt {1-2 x}}{1176 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 \sqrt {1-2 x}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {14101 \sqrt {1-2 x}}{392 (2+3 x) \sqrt {3+5 x}}+\frac {1463447 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.55 \begin {gather*} \frac {1463447 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {7 \sqrt {1-2 x} \left (5756085 x^3+11385261 x^2+7502166 x+1646704\right )}{(3 x+2)^3 \sqrt {5 x+3}}}{2744} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

((-7*Sqrt[1 - 2*x]*(1646704 + 7502166*x + 11385261*x^2 + 5756085*x^3))/((2 + 3*x)^3*Sqrt[3 + 5*x]) + 1463447*S
qrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2744

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IntegrateAlgebraic [A]  time = 0.26, size = 122, normalized size = 0.85 \begin {gather*} \frac {1463447 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}}-\frac {\sqrt {1-2 x} \left (\frac {98000 (1-2 x)^3}{(5 x+3)^3}+\frac {3220953 (1-2 x)^2}{(5 x+3)^2}+\frac {27317976 (1-2 x)}{5 x+3}[phone]\right )}{392 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

-1/392*(Sqrt[1 - 2*x]*(71708903 + (98000*(1 - 2*x)^3)/(3 + 5*x)^3 + (3220953*(1 - 2*x)^2)/(3 + 5*x)^2 + (27317
976*(1 - 2*x))/(3 + 5*x)))/(Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^3) + (1463447*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7
]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

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fricas [A]  time = 1.30, size = 116, normalized size = 0.81 \begin {gather*} \frac {1463447 \, \sqrt {7} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 (5756085 \, x^{3} + 11385261 \, x^{2} + 7502166 \, x + 1646704\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5488 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/5488*(1463447*sqrt(7)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(5756085*x^3 + 11385261*x^2 + 7502166*x + 1646704)*sqrt(5*x + 3)*sqrt
(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)

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giac [B]  time = 2.55, size = 374, normalized size = 2.60 \begin {gather*} -\frac {1}{54880} \, \sqrt {5} {\left (1463447 \, \sqrt {70} \sqrt {2} {\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 )} + 686000 \, \sqrt {2} {\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 {27720 \, \sqrt {2} {\left (11747 \, {\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 )}^{5} + 5216960 \, {\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 {615675200 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {2462700800 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\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 )}^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-1/54880*sqrt(5)*(1463447*sqrt(70)*sqrt(2)*(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)))) + 686000*sqrt(2)*((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))) + 27720*sqrt(2)*(11747*
((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)))^5
+ 5216960*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))^3 + 615675200*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 2462700800*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^2 + 280)^3)

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maple [B]  time = 0.02, size = 250, normalized size = 1.74 \begin {gather*} -\frac {\left (197565345 \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}[phone] \sqrt {7}\, x \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 [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 ) \sqrt {-2 x +1}}{5488 \left (3 x +2\right )^{3} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(1/2)/(3*x+2)^4/(5*x+3)^(3/2),x)

[Out]

-1/5488*(197565345*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+216590156*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+35122728*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+105030
324*(-10*x^2-x+3)^(1/2)*x+23053856*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(3*x+2)^3/(-10*x^2-x+3)^(1/2)/(5*x+3)^(
1/2)

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maxima [A]  time = 1.27, size = 211, normalized size = 1.47 \begin {gather*} -\frac {1463447}{5488} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {639565 \, x}{588 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {222589}{392 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7}{9 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {235}{36 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {13777}{168 \, {\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((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-1463447/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 639565/588*x/sqrt(-10*x^2 - x + 3) -
222589/392/sqrt(-10*x^2 - x + 3) + 7/9/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt
(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 235/36/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3
)*x + 4*sqrt(-10*x^2 - x + 3)) + 13777/168/(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 {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)/((3*x + 2)^4*(5*x + 3)^(3/2)),x)

[Out]

int((1 - 2*x)^(1/2)/((3*x + 2)^4*(5*x + 3)^(3/2)), 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)**(1/2)/(2+3*x)**4/(3+5*x)**(3/2),x)

[Out]

Timed out

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