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

Optimal. Leaf size=67 \[ \frac {164 \sqrt {5 x+3}}{3993 \sqrt {1-2 x}}+\frac {82 \sqrt {5 x+3}}{1815 (1-2 x)^{3/2}}-\frac {2}{55 (1-2 x)^{3/2} \sqrt {5 x+3}} \]

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Rubi [A]  time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \begin {gather*} \frac {164 \sqrt {5 x+3}}{3993 \sqrt {1-2 x}}+\frac {82 \sqrt {5 x+3}}{1815 (1-2 x)^{3/2}}-\frac {2}{55 (1-2 x)^{3/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(55*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (82*Sqrt[3 + 5*x])/(1815*(1 - 2*x)^(3/2)) + (164*Sqrt[3 + 5*x])/(3993*
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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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} (3+5 x)^{3/2}} \, dx &=-\frac {2}{55 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {41}{55} \int \frac {1}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2}{55 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {82 \sqrt {3+5 x}}{1815 (1-2 x)^{3/2}}+\frac {82}{363} \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2}{55 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {82 \sqrt {3+5 x}}{1815 (1-2 x)^{3/2}}+\frac {164 \sqrt {3+5 x}}{3993 \sqrt {1-2 x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 0.48 \begin {gather*} \frac {-1640 x^2+738 x+888}{3993 (1-2 x)^{3/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(888 + 738*x - 1640*x^2)/(3993*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])

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IntegrateAlgebraic [A]  time = 0.09, size = 54, normalized size = 0.81 \begin {gather*} -\frac {2 (5 x+3)^{3/2} \left (\frac {15 (1-2 x)^2}{(5 x+3)^2}-\frac {111 (1-2 x)}{5 x+3}-14\right )}{3993 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 5*x)^(3/2)*(-14 + (15*(1 - 2*x)^2)/(3 + 5*x)^2 - (111*(1 - 2*x))/(3 + 5*x)))/(3993*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.36, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (820 \, x^{2} - 369 \, x - 444\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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)^(3/2),x, algorithm="fricas")

[Out]

-2/3993*(820*x^2 - 369*x - 444)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(20*x^3 - 8*x^2 - 7*x + 3)

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giac [B]  time = 1.11, size = 100, normalized size = 1.49 \begin {gather*} -\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{2662 \, \sqrt {5 \, x + 3}} - \frac {2 \, {\left (152 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1221 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{99825 \, {\left (2 \, x - 1\right )}^{2}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{1331 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\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)^(3/2),x, algorithm="giac")

[Out]

-1/2662*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 2/99825*(152*sqrt(5)*(5*x + 3) - 1221*sq
rt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/1331*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))

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maple [A]  time = 0.00, size = 27, normalized size = 0.40 \begin {gather*} -\frac {2 \left (820 x^{2}-369 x -444\right )}{3993 \sqrt {5 x +3}\, \left (-2 x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3993*(820*x^2-369*x-444)/(5*x+3)^(1/2)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.55, size = 64, normalized size = 0.96 \begin {gather*} \frac {820 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {41}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {7}{33 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\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)^(3/2),x, algorithm="maxima")

[Out]

820/3993*x/sqrt(-10*x^2 - x + 3) + 41/3993/sqrt(-10*x^2 - x + 3) - 7/33/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*
x^2 - x + 3))

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mupad [B]  time = 2.47, size = 52, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {5\,x+3}\,\left (-\frac {164\,x^2}{3993}+\frac {123\,x}{6655}+\frac {148}{6655}\right )}{\frac {x\,\sqrt {1-2\,x}}{10}-\frac {3\,\sqrt {1-2\,x}}{10}+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)^(5/2)*(5*x + 3)^(3/2)),x)

[Out]

-((5*x + 3)^(1/2)*((123*x)/6655 - (164*x^2)/3993 + 148/6655))/((x*(1 - 2*x)^(1/2))/10 - (3*(1 - 2*x)^(1/2))/10
+ 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}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, 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)**(3/2),x)

[Out]

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

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