3.26.10 \(\int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\) [2510]

Optimal. Leaf size=74 \[ -\frac {2 \sqrt {1-2 x}}{825 (3+5 x)^{3/2}}-\frac {404 \sqrt {1-2 x}}{9075 \sqrt {3+5 x}}+\frac {9}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]

[Out]

9/125*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/825*(1-2*x)^(1/2)/(3+5*x)^(3/2)-404/9075*(1-2*x)^(1/2)/(3
+5*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 74, 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 = {91, 79, 56, 222} \begin {gather*} \frac {9}{25} \sqrt {\frac {2}{5}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {404 \sqrt {1-2 x}}{9075 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{825 (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(825*(3 + 5*x)^(3/2)) - (404*Sqrt[1 - 2*x])/(9075*Sqrt[3 + 5*x]) + (9*Sqrt[2/5]*ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]])/25

Rule 56

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 79

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] || L
tQ[p, n]))))

Rule 91

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 222

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}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{825 (3+5 x)^{3/2}}+\frac {2}{825} \int \frac {\frac {1093}{2}+\frac {1485 x}{2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{825 (3+5 x)^{3/2}}-\frac {404 \sqrt {1-2 x}}{9075 \sqrt {3+5 x}}+\frac {9}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{825 (3+5 x)^{3/2}}-\frac {404 \sqrt {1-2 x}}{9075 \sqrt {3+5 x}}+\frac {18 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x}}{825 (3+5 x)^{3/2}}-\frac {404 \sqrt {1-2 x}}{9075 \sqrt {3+5 x}}+\frac {9}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 61, normalized size = 0.82 \begin {gather*} -\frac {2 \sqrt {1-2 x} (617+1010 x)}{9075 (3+5 x)^{3/2}}-\frac {9}{25} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(617 + 1010*x))/(9075*(3 + 5*x)^(3/2)) - (9*Sqrt[2/5]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])
/25

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Maple [A]
time = 0.08, size = 96, normalized size = 1.30

method	result	size
default	\(\frac {\left (81675 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+98010 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +29403 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-20200 x \sqrt {-10 x^{2}-x +3}-12340 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{90750 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90750*(81675*10^(1/2)*arcsin(20/11*x+1/11)*x^2+98010*10^(1/2)*arcsin(20/11*x+1/11)*x+29403*10^(1/2)*arcsin(2
0/11*x+1/11)-20200*x*(-10*x^2-x+3)^(1/2)-12340*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^
(3/2)

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Maxima [A]
time = 0.54, size = 62, normalized size = 0.84 \begin {gather*} \frac {9}{250} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{825 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {404 \, \sqrt {-10 \, x^{2} - x + 3}}{9075 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/250*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 2/825*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 404/9075*sqrt
(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]
time = 0.42, size = 92, normalized size = 1.24 \begin {gather*} -\frac {3267 \, \sqrt {5} \sqrt {2} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (1010 \, x + 617\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{90750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90750*(3267*sqrt(5)*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-
2*x + 1)/(10*x^2 + x - 3)) + 20*(1010*x + 617)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (51) = 102\).
time = 1.38, size = 139, normalized size = 1.88 \begin {gather*} -\frac {1}{726000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {1620 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {9}{125} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {405 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{45375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/726000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 1620*(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3)) + 9/125*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/45375*sqrt(10)*(5*x + 3)^(3/2
)*(405*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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