3.2564 \(\int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ -\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {5 x+3}}+\frac {49}{22 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {9 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}} \]

[Out]

-9/50*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+49/22/(1-2*x)^(1/2)/(3+5*x)^(1/2)-1229/1210*(1-2*x)^(1/2)/(
3+5*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 72, 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 = {89, 78, 54, 216} \[ -\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {5 x+3}}+\frac {49}{22 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {9 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(22*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (1229*Sqrt[1 - 2*x])/(1210*Sqrt[3 + 5*x]) - (9*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(5*Sqrt[10])

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 78

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{22} \int \frac {-\frac {127}{2}+99 x}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {3+5 x}}-\frac {9}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {3+5 x}}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5 \sqrt {5}}\\ &=\frac {49}{22 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1229 \sqrt {1-2 x}}{1210 \sqrt {3+5 x}}-\frac {9 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 77, normalized size = 1.07 \[ \frac {10 \sqrt {2 x-1} (1229 x+733)-1089 (2 x-1) \sqrt {50 x+30} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{6050 \sqrt {-(1-2 x)^2} \sqrt {5 x+3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(10*Sqrt[-1 + 2*x]*(733 + 1229*x) - 1089*(-1 + 2*x)*Sqrt[30 + 50*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(6050*
Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x])

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fricas [A]  time = 0.84, size = 82, normalized size = 1.14 \[ \frac {1089 \, \sqrt {10} {\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (1229 \, x + 733\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12100 \, {\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12100*(1089*sqrt(10)*(10*x^2 + x - 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 20*(1229*x + 733)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(10*x^2 + x - 3)

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giac [B]  time = 1.25, size = 105, normalized size = 1.46 \[ -\frac {9}{50} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{6050 \, \sqrt {5 \, x + 3}} - \frac {49 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{605 \, {\left (2 \, x - 1\right )}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{3025 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/50*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/6050*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt
(5*x + 3) - 49/605*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 2/3025*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))

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maple [A]  time = 0.02, size = 103, normalized size = 1.43 \[ -\frac {\sqrt {-2 x +1}\, \left (10890 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1089 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+24580 \sqrt {-10 x^{2}-x +3}\, x -3267 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+14660 \sqrt {-10 x^{2}-x +3}\right )}{12100 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12100*(-2*x+1)^(1/2)*(10890*10^(1/2)*x^2*arcsin(20/11*x+1/11)+1089*10^(1/2)*x*arcsin(20/11*x+1/11)-3267*10^
(1/2)*arcsin(20/11*x+1/11)+24580*(-10*x^2-x+3)^(1/2)*x+14660*(-10*x^2-x+3)^(1/2))/(2*x-1)/(-10*x^2-x+3)^(1/2)/
(5*x+3)^(1/2)

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maxima [A]  time = 1.09, size = 41, normalized size = 0.57 \[ \frac {9}{100} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1229 \, x}{605 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {733}{605 \, \sqrt {-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/100*sqrt(10)*arcsin(-20/11*x - 1/11) + 1229/605*x/sqrt(-10*x^2 - x + 3) + 733/605/sqrt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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