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

Optimal. Leaf size=116 \[ -\frac {938 (5 x+3)^{5/2}}{363 \sqrt {1-2 x}}+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {40787 \sqrt {1-2 x} (5 x+3)^{3/2}}{5808}-\frac {40787}{704} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {40787 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64 \sqrt {10}} \]

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Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {89, 78, 50, 54, 216} \begin {gather*} -\frac {938 (5 x+3)^{5/2}}{363 \sqrt {1-2 x}}+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {40787 \sqrt {1-2 x} (5 x+3)^{3/2}}{5808}-\frac {40787}{704} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {40787 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-40787*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/704 - (40787*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/5808 + (49*(3 + 5*x)^(5/2))/(
66*(1 - 2*x)^(3/2)) - (938*(3 + 5*x)^(5/2))/(363*Sqrt[1 - 2*x]) + (40787*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64
*Sqrt[10])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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])))

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {(3+5 x)^{3/2} \left (\frac {1579}{2}+297 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{1452}\\ &=-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787}{352} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {40787}{704} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787}{128} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {40787}{704} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{64 \sqrt {5}}\\ &=-\frac {40787}{704} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 90, normalized size = 0.78 \begin {gather*} \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (2160 x^3+12780 x^2-52256 x+18351\right )+122361 \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1920 \sqrt {1-2 x} (2 x-1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(18351 - 52256*x + 12780*x^2 + 2160*x^3) + 122361*Sqrt[10]*(1 - 2*x)^2*ArcSin
h[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(1920*Sqrt[1 - 2*x]*(-1 + 2*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.18, size = 125, normalized size = 1.08 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {611805 (1-2 x)^3}{(5 x+3)^3}-\frac {407870 (1-2 x)^2}{(5 x+3)^2}-\frac {52192 (1-2 x)}{5 x+3}+3136\right )}{192 (1-2 x)^{3/2} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {40787 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{64 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 5*x)^(3/2)*(3136 - (611805*(1 - 2*x)^3)/(3 + 5*x)^3 - (407870*(1 - 2*x)^2)/(3 + 5*x)^2 - (52192*(1 - 2*x
))/(3 + 5*x)))/(192*(1 - 2*x)^(3/2)*(2 + (5*(1 - 2*x))/(3 + 5*x))^2) - (40787*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])
/Sqrt[3 + 5*x]])/(64*Sqrt[10])

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fricas [A]  time = 1.26, size = 96, normalized size = 0.83 \begin {gather*} -\frac {122361 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2160 \, x^{3} + 12780 \, x^{2} - 52256 \, x + 18351\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3840 \, {\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)^2*(3+5*x)^(3/2)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3840*(122361*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) + 20*(2160*x^3 + 12780*x^2 - 52256*x + 18351)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.08, size = 84, normalized size = 0.72 \begin {gather*} \frac {40787}{640} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 247 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 81574 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1345971 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{24000 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

40787/640*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/24000*(4*(9*(12*sqrt(5)*(5*x + 3) + 247*sqrt(5))*(5
*x + 3) - 81574*sqrt(5))*(5*x + 3) + 1345971*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.02, size = 137, normalized size = 1.18 \begin {gather*} \frac {\left (-43200 \sqrt {-10 x^{2}-x +3}\, x^{3}+489444 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-255600 \sqrt {-10 x^{2}-x +3}\, x^{2}-489444 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1045120 \sqrt {-10 x^{2}-x +3}\, x +122361 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-367020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{3840 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(489444*10^(1/2)*x^2*arcsin(20/11*x+1/11)-43200*(-10*x^2-x+3)^(1/2)*x^3-489444*10^(1/2)*x*arcsin(20/11*
x+1/11)-255600*(-10*x^2-x+3)^(1/2)*x^2+122361*10^(1/2)*arcsin(20/11*x+1/11)+1045120*(-10*x^2-x+3)^(1/2)*x-3670
20*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.36, size = 154, normalized size = 1.33 \begin {gather*} \frac {40787}{1280} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {297}{64} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {49 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{24 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {21 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (2 \, x - 1\right )}} + \frac {539 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {5873 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

40787/1280*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 297/64*sqrt(-10*x^2 - x + 3) - 49/24*(-10*x^2 - x + 3)^(3/
2)/(8*x^3 - 12*x^2 + 6*x - 1) + 21/4*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 9/16*(-10*x^2 - x + 3)^(3/2)/
(2*x - 1) + 539/48*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 5873/48*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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