3.2612 \(\int \frac {2+3 x}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=45 \[ \frac {7 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {29 \sqrt {5 x+3}}{363 \sqrt {1-2 x}} \]

[Out]

7/33*(3+5*x)^(1/2)/(1-2*x)^(3/2)-29/363*(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {78, 37} \[ \frac {7 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {29 \sqrt {5 x+3}}{363 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (29*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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]))))

Rubi steps

\begin {align*} \int \frac {2+3 x}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {29}{66} \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {29 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.60 \[ \frac {2 \sqrt {5 x+3} (29 x+24)}{363 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x]*(24 + 29*x))/(363*(1 - 2*x)^(3/2))

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fricas [A]  time = 0.63, size = 33, normalized size = 0.73 \[ \frac {2 \, {\left (29 \, x + 24\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/363*(29*x + 24)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.12, size = 39, normalized size = 0.87 \[ \frac {2 \, {\left (29 \, \sqrt {5} {\left (5 \, x + 3\right )} + 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{9075 \, {\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9075*(29*sqrt(5)*(5*x + 3) + 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.00, size = 22, normalized size = 0.49 \[ \frac {2 \sqrt {5 x +3}\, \left (29 x +24\right )}{363 \left (-2 x +1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/363*(5*x+3)^(1/2)*(29*x+24)/(-2*x+1)^(3/2)

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maxima [A]  time = 1.08, size = 48, normalized size = 1.07 \[ \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{33 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {29 \, \sqrt {-10 \, x^{2} - x + 3}}{363 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/33*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 29/363*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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mupad [B]  time = 2.43, size = 35, normalized size = 0.78 \[ -\frac {\sqrt {5\,x+3}\,\left (\frac {29\,x}{363}+\frac {8}{121}\right )}{x\,\sqrt {1-2\,x}-\frac {\sqrt {1-2\,x}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((5*x + 3)^(1/2)*((29*x)/363 + 8/121))/(x*(1 - 2*x)^(1/2) - (1 - 2*x)^(1/2)/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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