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

Optimal. Leaf size=67 \[ -\frac {4091 \sqrt {1-2 x}}{19965 \sqrt {5 x+3}}-\frac {3679 \sqrt {1-2 x}}{3630 (5 x+3)^{3/2}}+\frac {49}{22 (5 x+3)^{3/2} \sqrt {1-2 x}} \]

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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 = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {89, 78, 37} \begin {gather*} -\frac {4091 \sqrt {1-2 x}}{19965 \sqrt {5 x+3}}-\frac {3679 \sqrt {1-2 x}}{3630 (5 x+3)^{3/2}}+\frac {49}{22 (5 x+3)^{3/2} \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(3630*(3 + 5*x)^(3/2)) - (4091*Sqrt[1 - 2*x])/(19
965*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]))))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {49}{22 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1}{22} \int \frac {-\frac {617}{2}+99 x}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {49}{22 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {3679 \sqrt {1-2 x}}{3630 (3+5 x)^{3/2}}+\frac {4091 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{3630}\\ &=\frac {49}{22 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {3679 \sqrt {1-2 x}}{3630 (3+5 x)^{3/2}}-\frac {4091 \sqrt {1-2 x}}{19965 \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 (4091 x^2+4456 x+1196\right )}{3993 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1196 + 4456*x + 4091*x^2))/(3993*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.10, size = 53, normalized size = 0.79 \begin {gather*} -\frac {2 \sqrt {5 x+3} \left (\frac {(1-2 x)^2}{(5 x+3)^2}+\frac {42 (1-2 x)}{5 x+3}-147\right )}{3993 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 5*x]*(-147 + (1 - 2*x)^2/(3 + 5*x)^2 + (42*(1 - 2*x))/(3 + 5*x)))/(3993*Sqrt[1 - 2*x])

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fricas [A]  time = 1.55, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (4091 \, x^{2} + 4456 \, x + 1196\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)^2/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-2/3993*(4091*x^2 + 4456*x + 1196)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(50*x^3 + 35*x^2 - 12*x - 9)

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giac [B]  time = 1.26, size = 147, normalized size = 2.19 \begin {gather*} -\frac {1}{1597200} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {1668 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {98 \, \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 {417 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{99825 \, {\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)^2/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/1597200*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 1668*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3)) - 98/6655*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/99825*sqrt(10)*(5*x + 3
)^(3/2)*(417*(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 {8182}{3993} x^{2}+\frac {8912}{3993} x +\frac {2392}{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/(-2*x+1)^(3/2)/(5*x+3)^(5/2),x)

[Out]

2/3993*(4091*x^2+4456*x+1196)/(5*x+3)^(3/2)/(-2*x+1)^(1/2)

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maxima [A]  time = 0.56, size = 64, normalized size = 0.96 \begin {gather*} \frac {8182 \, x}{19965 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {20014}{99825 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{825 \, {\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)^2/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

8182/19965*x/sqrt(-10*x^2 - x + 3) + 20014/99825/sqrt(-10*x^2 - x + 3) - 2/825/(5*sqrt(-10*x^2 - x + 3)*x + 3*
sqrt(-10*x^2 - x + 3))

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mupad [B]  time = 0.29, size = 51, normalized size = 0.76 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {8182\,x^2}{99825}+\frac {8912\,x}{99825}+\frac {2392}{99825}\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)^2/((1 - 2*x)^(3/2)*(5*x + 3)^(5/2)),x)

[Out]

((5*x + 3)^(1/2)*((8912*x)/99825 + (8182*x^2)/99825 + 2392/99825))/((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(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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