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

Optimal. Leaf size=93 \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)}-\frac {121 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]

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Rubi [A]  time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac {11 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)}-\frac {121 \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]*Sqrt[3 + 5*x])/(2 + 3*x)^3,x]

[Out]

(-11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2*(2 + 3*x)^2) - (121*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*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 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} \sqrt {3+5 x}}{(2+3 x)^3} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {11}{4} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {121}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac {121}{28} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}-\frac {121 \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 = 0.04, size = 69, normalized size = 0.74 \begin {gather*} \frac {1}{196} \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (37 x+20)}{(3 x+2)^2}-121 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(20 + 37*x))/(2 + 3*x)^2 - 121*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/196

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IntegrateAlgebraic [A]  time = 0.18, size = 89, normalized size = 0.96 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}-7\right )}{28 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^2}-\frac {121 \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]*Sqrt[3 + 5*x])/(2 + 3*x)^3,x]

[Out]

(-121*Sqrt[1 - 2*x]*(-7 + (1 - 2*x)/(3 + 5*x)))/(28*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^2) - (121*ArcTan[S
qrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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fricas [A]  time = 1.38, size = 86, normalized size = 0.92 \begin {gather*} -\frac {121 \, \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 (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/392*(121*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*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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giac [B]  time = 1.51, size = 250, normalized size = 2.69 \begin {gather*} \frac {121}{3920} \, \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 {121 \, \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 {280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{14 \, {\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((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3,x, algorithm="giac")

[Out]

121/3920*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)))) - 121/14*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 - 280*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) + 1120*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 = 0.01, size = 154, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (1089 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1452 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+518 \sqrt {-10 x^{2}-x +3}\, x +484 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+280 \sqrt {-10 x^{2}-x +3}\right )}{392 \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((-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)^3,x)

[Out]

1/392*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(1089*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1452*7
^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+484*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))+518*(-10*x^2-x+3)^(1/2)*x+280*(-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 = 1.46, size = 90, normalized size = 0.97 \begin {gather*} \frac {121}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {5}{21} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {37 \, \sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 5/21*sqrt(-10*x^2 - x + 3) + 3/14*(-10*x^2
- x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 37/84*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [B]  time = 12.62, size = 1037, normalized size = 11.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2129*((1 - 2*x)^(1/2) - 1)^5)/(875*(3^(1/2) - (5*x + 3)^(1/2))^5) - (4258*((1 - 2*x)^(1/2) - 1)^3)/(4375*(3^
(1/2) - (5*x + 3)^(1/2))^3) - (158*((1 - 2*x)^(1/2) - 1))/(4375*(3^(1/2) - (5*x + 3)^(1/2))) + (79*((1 - 2*x)^
(1/2) - 1)^7)/(140*(3^(1/2) - (5*x + 3)^(1/2))^7) + (991*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(4375*(3^(1/2) - (5*
x + 3)^(1/2))^2) - (376*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(875*(3^(1/2) - (5*x + 3)^(1/2))^4) + (991*3^(1/2)*((
1 - 2*x)^(1/2) - 1)^6)/(700*(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 - (96*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) - (121*7^(1/2)*atan(((121*7^(1/2)*((7
26*3^(1/2))/875 + (363*((1 - 2*x)^(1/2) - 1))/(875*(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)*121i)/392 - (363*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(175*(3^(1/2) - (5*x + 3)^(1/2))^2)))/39
2 + (121*7^(1/2)*((726*3^(1/2))/875 + (363*((1 - 2*x)^(1/2) - 1))/(875*(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)*121i)/392 - (363*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(175*(3^(1/2) - (5*x
+ 3)^(1/2))^2)))/392)/((7^(1/2)*((726*3^(1/2))/875 + (363*((1 - 2*x)^(1/2) - 1))/(875*(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)*121i)/392 - (363*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(175
*(3^(1/2) - (5*x + 3)^(1/2))^2))*121i)/392 - (7^(1/2)*((726*3^(1/2))/875 + (363*((1 - 2*x)^(1/2) - 1))/(875*(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)*121i)/392 - (363*3^(1/2)*((1 - 2*
x)^(1/2) - 1)^2)/(175*(3^(1/2) - (5*x + 3)^(1/2))^2))*121i)/392 + (14641*((1 - 2*x)^(1/2) - 1)^2)/(9800*(3^(1/
2) - (5*x + 3)^(1/2))^2) + 14641/24500)))/196

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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