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

Optimal. Leaf size=45 \[ -\frac {206 \sqrt {1-2 x}}{1815 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{165 (5 x+3)^{3/2}} \]

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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} \begin {gather*} -\frac {206 \sqrt {1-2 x}}{1815 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{165 (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(165*(3 + 5*x)^(3/2)) - (206*Sqrt[1 - 2*x])/(1815*Sqrt[3 + 5*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}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{165 (3+5 x)^{3/2}}+\frac {103}{165} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{165 (3+5 x)^{3/2}}-\frac {206 \sqrt {1-2 x}}{1815 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.60 \begin {gather*} -\frac {2 \sqrt {1-2 x} (103 x+64)}{363 (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(64 + 103*x))/(363*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.07, size = 37, normalized size = 0.82 \begin {gather*} -\frac {2 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}+21\right )}{363 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(21 + (1 - 2*x)/(3 + 5*x)))/(363*Sqrt[3 + 5*x])

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fricas [A]  time = 1.35, size = 33, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (103 \, x + 64\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\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)/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-2/363*(103*x + 64)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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giac [B]  time = 0.96, size = 121, normalized size = 2.69 \begin {gather*} -\frac {1}{145200} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {828 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {207 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{9075 \, {\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)/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/145200*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 828*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3)) + 1/9075*sqrt(10)*(5*x + 3)^(3/2)*(207*(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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maple [A]  time = 0.00, size = 22, normalized size = 0.49 \begin {gather*} -\frac {2 \left (103 x +64\right ) \sqrt {-2 x +1}}{363 \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(103*x+64)/(5*x+3)^(3/2)*(-2*x+1)^(1/2)

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maxima [A]  time = 1.51, size = 48, normalized size = 1.07 \begin {gather*} -\frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{165 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {206 \, \sqrt {-10 \, x^{2} - x + 3}}{1815 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/165*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 206/1815*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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mupad [B]  time = 2.50, size = 51, normalized size = 1.13 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {412\,x^2}{9075}+\frac {2\,x}{363}-\frac {128}{9075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{5}+\frac {9\,\sqrt {1-2\,x}}{25}+x^2\,\sqrt {1-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((2*x)/363 + (412*x^2)/9075 - 128/9075))/((6*x*(1 - 2*x)^(1/2))/5 + (9*(1 - 2*x)^(1/2))/25 +
x^2*(1 - 2*x)^(1/2))

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

[Out]

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

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