3.27.12 \(\int \frac {2+3 x}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\) [2612]

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

[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 = 0.00, 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 = {79, 37} \begin {gather*} \frac {7 \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {29 \sqrt {5 x+3}}{363 \sqrt {1-2 x}} \end {gather*}

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 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]))))

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 = 0.06, size = 27, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {3+5 x} (24+29 x)}{363 (1-2 x)^{3/2}} \end {gather*}

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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Maple [A]
time = 0.08, size = 29, normalized size = 0.64

method	result	size
gosper	\(\frac {2 \sqrt {3+5 x}\, \left (29 x +24\right )}{363 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(22\)
default	\(\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (29 x +24\right )}{363 \left (-1+2 x \right )^{2}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 48, normalized size = 1.07 \begin {gather*} \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 )}} \end {gather*}

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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Fricas [A]
time = 0.45, size = 33, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (29 \, x + 24\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

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

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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Giac [A]
time = 3.36, size = 39, normalized size = 0.87 \begin {gather*} \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}} \end {gather*}

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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Mupad [B]
time = 2.43, size = 35, normalized size = 0.78 \begin {gather*} -\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}} \end {gather*}

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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