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

Optimal. Leaf size=118 \[ \frac {7 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}+\frac {81}{44} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {405}{32} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {13365}{128} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {29403}{128} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {78, 50, 54, 216} \begin {gather*} \frac {7 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}+\frac {81}{44} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {405}{32} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {13365}{128} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {29403}{128} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(13365*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/128 + (405*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/32 + (81*Sqrt[1 - 2*x]*(3 + 5*x)
^(5/2))/44 + (7*(3 + 5*x)^(7/2))/(11*Sqrt[1 - 2*x]) - (29403*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/128

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 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) (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {243}{22} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {405}{8} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {13365}{64} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {13365}{128} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {147015}{256} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {13365}{128} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {1}{128} \left (29403 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {13365}{128} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {405}{32} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {81}{44} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {7 (3+5 x)^{7/2}}{11 \sqrt {1-2 x}}-\frac {29403}{128} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 83, normalized size = 0.70 \begin {gather*} \frac {-2 \sqrt {2 x-1} \sqrt {5 x+3} \left (1600 x^3+6120 x^2+14526 x-22545\right )-29403 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{256 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-22545 + 14526*x + 6120*x^2 + 1600*x^3) - 29403*Sqrt[10]*(-1 + 2*x)*ArcSinh[
Sqrt[5/11]*Sqrt[-1 + 2*x]])/(256*Sqrt[-(1 - 2*x)^2])

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IntegrateAlgebraic [A]  time = 0.18, size = 127, normalized size = 1.08 \begin {gather*} \frac {121 \sqrt {5 x+3} \left (\frac {30375 (1-2 x)^3}{(5 x+3)^3}+\frac {32400 (1-2 x)^2}{(5 x+3)^2}+\frac {10692 (1-2 x)}{5 x+3}+896\right )}{128 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}+\frac {29403}{128} \sqrt {\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(121*Sqrt[3 + 5*x]*(896 + (30375*(1 - 2*x)^3)/(3 + 5*x)^3 + (32400*(1 - 2*x)^2)/(3 + 5*x)^2 + (10692*(1 - 2*x)
)/(3 + 5*x)))/(128*Sqrt[1 - 2*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^3) + (29403*Sqrt[5/2]*ArcTan[(Sqrt[5/2]*Sqrt[1
- 2*x])/Sqrt[3 + 5*x]])/128

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fricas [A]  time = 1.25, size = 92, normalized size = 0.78 \begin {gather*} \frac {29403 \, \sqrt {5} \sqrt {2} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 4 \, {\left (1600 \, x^{3} + 6120 \, x^{2} + 14526 \, x - 22545\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{512 \, {\left (2 \, x - 1\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)^(3/2),x, algorithm="fricas")

[Out]

1/512*(29403*sqrt(5)*sqrt(2)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10
*x^2 + x - 3)) + 4*(1600*x^3 + 6120*x^2 + 14526*x - 22545)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 0.98, size = 84, normalized size = 0.71 \begin {gather*} -\frac {29403}{256} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 81 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4455 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 147015 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3200 \, {\left (2 \, x - 1\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)^(3/2),x, algorithm="giac")

[Out]

-29403/256*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/3200*(2*(4*(8*sqrt(5)*(5*x + 3) + 81*sqrt(5))*(5*x
+ 3) + 4455*sqrt(5))*(5*x + 3) - 147015*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.01, size = 123, normalized size = 1.04 \begin {gather*} -\frac {\left (-6400 \sqrt {-10 x^{2}-x +3}\, x^{3}-24480 \sqrt {-10 x^{2}-x +3}\, x^{2}+58806 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-58104 \sqrt {-10 x^{2}-x +3}\, x -29403 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+90180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{512 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \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)^(3/2),x)

[Out]

-1/512*(-6400*(-10*x^2-x+3)^(1/2)*x^3+58806*10^(1/2)*x*arcsin(20/11*x+1/11)-24480*(-10*x^2-x+3)^(1/2)*x^2-2940
3*10^(1/2)*arcsin(20/11*x+1/11)-58104*(-10*x^2-x+3)^(1/2)*x+90180*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^
(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.21, size = 92, normalized size = 0.78 \begin {gather*} -\frac {125 \, x^{4}}{2 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {4425 \, x^{3}}{16 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {45495 \, x^{2}}{64 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {29403}{512} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {69147 \, x}{128 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {67635}{128 \, \sqrt {-10 \, x^{2} - x + 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)^(3/2),x, algorithm="maxima")

[Out]

-125/2*x^4/sqrt(-10*x^2 - x + 3) - 4425/16*x^3/sqrt(-10*x^2 - x + 3) - 45495/64*x^2/sqrt(-10*x^2 - x + 3) + 29
403/512*sqrt(10)*arcsin(-20/11*x - 1/11) + 69147/128*x/sqrt(-10*x^2 - x + 3) + 67635/128/sqrt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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