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

Optimal. Leaf size=93 \[ \frac {3 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}-\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)}-\frac {451 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \]

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Rubi [A]  time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {3 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}-\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)}-\frac {451 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)) + (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(14*(2 + 3*x)^2) - (451*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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 94

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[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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 {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx &=\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac {41}{28} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac {451}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac {451}{196} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}-\frac {451 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 69, normalized size = 0.74 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (87 x+44)}{(3 x+2)^2}-451 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(44 + 87*x))/(2 + 3*x)^2 - 451*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/1372

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IntegrateAlgebraic [A]  time = 2.17, size = 176, normalized size = 1.89 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (87 \sqrt {5} (5 x+3)^{3/2}-41 \sqrt {5} \sqrt {5 x+3}\right )}{196 (3 (5 x+3)+1)^2}-\frac {451 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{196 \sqrt {7}}-\frac {451 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{196 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-41*Sqrt[5]*Sqrt[3 + 5*x] + 87*Sqrt[5]*(3 + 5*x)^(3/2)))/(196*(1 + 3*(3 + 5*x))^2) -
(451*ArcTan[(Sqrt[2/(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(196*Sqrt[7]) - (4
51*ArcTan[(Sqrt[68 + 2*Sqrt[1155]]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(196*Sqrt[7])

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fricas [A]  time = 1.23, size = 86, normalized size = 0.92 \begin {gather*} -\frac {451 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\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 (87 \, x + 44\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2744 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2744*(451*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 14*(87*x + 44)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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giac [B]  time = 2.02, size = 252, normalized size = 2.71 \begin {gather*} \frac {451}{27440} \, \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 {11 \, \sqrt {10} {\left (41 \, {\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 {7000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {28000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

451/27440*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)))) - 11/98*sqrt(10)*(41*((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 - 7000*(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))/sqrt(5*x + 3) + 28000*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)^2

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maple [B]  time = 0.02, size = 154, normalized size = 1.66 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (4059 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5412 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1218 \sqrt {-10 x^{2}-x +3}\, x +1804 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+616 \sqrt {-10 x^{2}-x +3}\right )}{2744 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2744*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(4059*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+5412*
7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1804*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))+1218*(-10*x^2-x+3)^(1/2)*x+616*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^2

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maxima [A]  time = 1.18, size = 76, normalized size = 0.82 \begin {gather*} \frac {451}{2744} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {29 \, \sqrt {-10 \, x^{2} - x + 3}}{196 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

451/2744*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/14*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x
+ 4) + 29/196*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [B]  time = 13.13, size = 1037, normalized size = 11.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((199*((1 - 2*x)^(1/2) - 1)^5)/(245*(3^(1/2) - (5*x + 3)^(1/2))^5) - (398*((1 - 2*x)^(1/2) - 1)^3)/(1225*(3^(1
/2) - (5*x + 3)^(1/2))^3) - (314*((1 - 2*x)^(1/2) - 1))/(30625*(3^(1/2) - (5*x + 3)^(1/2))) + (157*((1 - 2*x)^
(1/2) - 1)^7)/(980*(3^(1/2) - (5*x + 3)^(1/2))^7) + (2197*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(30625*(3^(1/2) - (
5*x + 3)^(1/2))^2) - (4276*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(30625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (2197*3^(1
/2)*((1 - 2*x)^(1/2) - 1)^6)/(4900*(3^(1/2) - (5*x + 3)^(1/2))^6))/((544*((1 - 2*x)^(1/2) - 1)^2)/(625*(3^(1/2
) - (5*x + 3)^(1/2))^2) - (1764*((1 - 2*x)^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (136*((1 - 2*x)
^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^(1/2) - (5*x + 3)^(1/2))^8 - (9
6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(625*(3^(1/2) - (5*x + 3)^(1/2))^3) + (48*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/
(125*(3^(1/2) - (5*x + 3)^(1/2))^5) + (12*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(5*(3^(1/2) - (5*x + 3)^(1/2))^7) -
(96*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(625*(3^(1/2) - (5*x + 3)^(1/2))) + 16/625) - (451*7^(1/2)*atan(((451*7^(1
/2)*((2706*3^(1/2))/6125 + (1353*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*((212*((
1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2)
- (5*x + 3)^(1/2))) - 536/125)*451i)/2744 - (1353*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^
(1/2))^2)))/2744 + (451*7^(1/2)*((2706*3^(1/2))/6125 + (1353*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)
^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x
)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*451i)/2744 - (1353*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)
/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2)))/2744)/((7^(1/2)*((2706*3^(1/2))/6125 + (1353*((1 - 2*x)^(1/2) - 1))/(6
125*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2)
+ (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*451i)/2744 - (1353*3^(1/2)*
((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2))*451i)/2744 - (7^(1/2)*((2706*3^(1/2))/6125 + (1
353*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3
^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125
)*451i)/2744 - (1353*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2))*451i)/2744 + (2034
01*((1 - 2*x)^(1/2) - 1)^2)/(480200*(3^(1/2) - (5*x + 3)^(1/2))^2) + 203401/1200500)))/1372

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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