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

Optimal. Leaf size=67 \[ -\frac {428 \sqrt {1-2 x}}{3993 \sqrt {5 x+3}}-\frac {107 \sqrt {1-2 x}}{363 (5 x+3)^{3/2}}+\frac {7}{11 (5 x+3)^{3/2} \sqrt {1-2 x}} \]

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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 {428 \sqrt {1-2 x}}{3993 \sqrt {5 x+3}}-\frac {107 \sqrt {1-2 x}}{363 (5 x+3)^{3/2}}+\frac {7}{11 (5 x+3)^{3/2} \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (107*Sqrt[1 - 2*x])/(363*(3 + 5*x)^(3/2)) - (428*Sqrt[1 - 2*x])/(3993*S
qrt[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 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)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {7}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {107}{22} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {7}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {107 \sqrt {1-2 x}}{363 (3+5 x)^{3/2}}+\frac {214}{363} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {7}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {107 \sqrt {1-2 x}}{363 (3+5 x)^{3/2}}-\frac {428 \sqrt {1-2 x}}{3993 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 0.48 \begin {gather*} \frac {2 \left (2140 x^2+1391 x+40\right )}{3993 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(40 + 1391*x + 2140*x^2))/(3993*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.09, size = 54, normalized size = 0.81 \begin {gather*} -\frac {2 \sqrt {5 x+3} \left (\frac {5 (1-2 x)^2}{(5 x+3)^2}+\frac {111 (1-2 x)}{5 x+3}-42\right )}{3993 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 5*x]*(-42 + (5*(1 - 2*x)^2)/(3 + 5*x)^2 + (111*(1 - 2*x))/(3 + 5*x)))/(3993*Sqrt[1 - 2*x])

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fricas [A]  time = 0.95, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (2140 \, x^{2} + 1391 \, x + 40\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3993*(2140*x^2 + 1391*x + 40)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(50*x^3 + 35*x^2 - 12*x - 9)

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giac [B]  time = 1.24, size = 147, normalized size = 2.19 \begin {gather*} -\frac {1}{319440} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {876 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {28 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{6655 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {219 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{19965 \, {\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)/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/319440*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 876*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3)) - 28/6655*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/19965*sqrt(10)*(5*x + 3)^
(3/2)*(219*(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 = 27, normalized size = 0.40 \begin {gather*} \frac {\frac {4280}{3993} x^{2}+\frac {2782}{3993} x +\frac {80}{3993}}{\left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3993*(2140*x^2+1391*x+40)/(5*x+3)^(3/2)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.61, size = 64, normalized size = 0.96 \begin {gather*} \frac {856 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {214}{19965 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{165 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \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)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

856/3993*x/sqrt(-10*x^2 - x + 3) + 214/19965/sqrt(-10*x^2 - x + 3) - 2/165/(5*sqrt(-10*x^2 - x + 3)*x + 3*sqrt
(-10*x^2 - x + 3))

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mupad [B]  time = 2.45, size = 51, normalized size = 0.76 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {856\,x^2}{19965}+\frac {2782\,x}{99825}+\frac {16}{19965}\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)^(3/2)*(5*x + 3)^(5/2)),x)

[Out]

((5*x + 3)^(1/2)*((2782*x)/99825 + (856*x^2)/19965 + 16/19965))/((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}{\left (1 - 2 x\right )^{\frac {3}{2}} \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)/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

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

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