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

Optimal. Leaf size=137 \[ \frac {608185 \sqrt {1-2 x}}{924 \sqrt {5 x+3}}-\frac {6095 \sqrt {1-2 x}}{84 (5 x+3)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {126513 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]

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Rubi [A]  time = 0.05, antiderivative size = 137, 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 {608185 \sqrt {1-2 x}}{924 \sqrt {5 x+3}}-\frac {6095 \sqrt {1-2 x}}{84 (5 x+3)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {126513 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6095*Sqrt[1 - 2*x])/(84*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (243*Sqrt[1 - 2*x
])/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (608185*Sqrt[1 - 2*x])/(924*Sqrt[3 + 5*x]) - (126513*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(28*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)^3 (3+5 x)^{5/2}} \, dx &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {1}{2} \int \frac {-\frac {41}{2}+30 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}-\frac {1}{14} \int \frac {-\frac {7577}{4}+2430 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {6095 \sqrt {1-2 x}}{84 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {1}{231} \int \frac {-\frac {855283}{8}+\frac {201135 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {6095 \sqrt {1-2 x}}{84 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {608185 \sqrt {1-2 x}}{924 \sqrt {3+5 x}}-\frac {2 \int -\frac {45924219}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2541}\\ &=-\frac {6095 \sqrt {1-2 x}}{84 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {608185 \sqrt {1-2 x}}{924 \sqrt {3+5 x}}+\frac {126513}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {6095 \sqrt {1-2 x}}{84 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {608185 \sqrt {1-2 x}}{924 \sqrt {3+5 x}}+\frac {126513}{28} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {6095 \sqrt {1-2 x}}{84 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {608185 \sqrt {1-2 x}}{924 \sqrt {3+5 x}}-\frac {126513 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 79, normalized size = 0.58 \begin {gather*} \frac {\sqrt {1-2 x} \left (27368325 x^3+52308690 x^2+33277877 x+7046540\right )}{924 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {126513 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(7046540 + 33277877*x + 52308690*x^2 + 27368325*x^3))/(924*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (1265
13*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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IntegrateAlgebraic [A]  time = 0.21, size = 137, normalized size = 1.00 \begin {gather*} \frac {-\frac {7000 (1-2 x)^{7/2}}{(5 x+3)^{7/2}}+\frac {317800 (1-2 x)^{5/2}}{(5 x+3)^{5/2}}+\frac {6958151 (1-2 x)^{3/2}}{(5 x+3)^{3/2}}+\frac {29224503 \sqrt {1-2 x}}{\sqrt {5 x+3}}}{924 \left (\frac {1-2 x}{5 x+3}+7\right )^2}-\frac {126513 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-7000*(1 - 2*x)^(7/2))/(3 + 5*x)^(7/2) + (317800*(1 - 2*x)^(5/2))/(3 + 5*x)^(5/2) + (6958151*(1 - 2*x)^(3/2)
)/(3 + 5*x)^(3/2) + (29224503*Sqrt[1 - 2*x])/Sqrt[3 + 5*x])/(924*(7 + (1 - 2*x)/(3 + 5*x))^2) - (126513*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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fricas [A]  time = 1.05, size = 116, normalized size = 0.85 \begin {gather*} -\frac {4174929 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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 (27368325 \, x^{3} + 52308690 \, x^{2} + 33277877 \, x + 7046540\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12936 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12936*(4174929*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(27368325*x^3 + 52308690*x^2 + 33277877*x + 7046540)*sqrt(5*x + 3)*
sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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giac [B]  time = 2.50, size = 377, normalized size = 2.75 \begin {gather*} -\frac {1}{129360} \, \sqrt {5} {\left (1225 \, \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 )}^{3} - 4174929 \, \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 )} - 2910600 \, \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 {2744280 \, \sqrt {2} {\left (151 \, {\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 {36120 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {144480 \, \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 )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/129360*sqrt(5)*(1225*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)))^3 - 4174929*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)))) - 2910600*sqrt(2)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 2744
280*sqrt(2)*(151*((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 + 36120*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 144480*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^2 + 280)^2)

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maple [B]  time = 0.02, size = 250, normalized size = 1.82 \begin {gather*} \frac {\left (939359025 \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}}{12936 \left (3 x +2\right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12936*(939359025*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*arcta
n(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+951883812*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+150297444*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+46
5890278*(-10*x^2-x+3)^(1/2)*x+98651560*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(3*x+2)^2/(-10*x^2-x+3)^(1/2)/(5*x+
3)^(3/2)

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maxima [A]  time = 1.45, size = 172, normalized size = 1.26 \begin {gather*} \frac {126513}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {608185 \, x}{462 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {635003}{924 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1985 \, x}{6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {49}{18 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1645}{36 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {6433}{36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

126513/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 608185/462*x/sqrt(-10*x^2 - x + 3) + 63
5003/924/sqrt(-10*x^2 - x + 3) + 1985/6*x/(-10*x^2 - x + 3)^(3/2) + 49/18/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*
(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 1645/36/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x
+ 3)^(3/2)) - 6433/36/(-10*x^2 - x + 3)^(3/2)

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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 )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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