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

Optimal. Leaf size=136 \[ \frac {1935}{67228 \sqrt {1-2 x}}-\frac {129}{2744 (1-2 x)^{3/2} (3 x+2)}+\frac {215}{9604 (1-2 x)^{3/2}}-\frac {129}{2744 (1-2 x)^{3/2} (3 x+2)^2}-\frac {43}{588 (1-2 x)^{3/2} (3 x+2)^3}+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac {1935 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228} \]

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Rubi [A]  time = 0.05, antiderivative size = 150, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \begin {gather*} -\frac {5805 \sqrt {1-2 x}}{134456 (3 x+2)}-\frac {1935 \sqrt {1-2 x}}{19208 (3 x+2)^2}-\frac {387 \sqrt {1-2 x}}{1372 (3 x+2)^3}+\frac {387}{686 \sqrt {1-2 x} (3 x+2)^3}+\frac {43}{294 (1-2 x)^{3/2} (3 x+2)^3}+\frac {1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac {1935 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(84*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + 43/(294*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 387/(686*Sqrt[1 - 2*x]*(2 + 3*x)^3
) - (387*Sqrt[1 - 2*x])/(1372*(2 + 3*x)^3) - (1935*Sqrt[1 - 2*x])/(19208*(2 + 3*x)^2) - (5805*Sqrt[1 - 2*x])/(
134456*(2 + 3*x)) - (1935*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/67228

Rule 51

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*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 206

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

Rubi steps

\begin {align*} \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{28} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{196} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}+\frac {1161}{196} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}+\frac {1935 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1372}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}+\frac {5805 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{19208}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}-\frac {5805 \sqrt {1-2 x}}{134456 (2+3 x)}+\frac {5805 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{134456}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}-\frac {5805 \sqrt {1-2 x}}{134456 (2+3 x)}-\frac {5805 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{134456}\\ &=\frac {1}{84 (1-2 x)^{3/2} (2+3 x)^4}+\frac {43}{294 (1-2 x)^{3/2} (2+3 x)^3}+\frac {387}{686 \sqrt {1-2 x} (2+3 x)^3}-\frac {387 \sqrt {1-2 x}}{1372 (2+3 x)^3}-\frac {1935 \sqrt {1-2 x}}{19208 (2+3 x)^2}-\frac {5805 \sqrt {1-2 x}}{134456 (2+3 x)}-\frac {1935 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 42, normalized size = 0.31 \begin {gather*} \frac {688 \, _2F_1\left (-\frac {3}{2},4;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+\frac {2401}{(3 x+2)^4}}{201684 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2401/(2 + 3*x)^4 + 688*Hypergeometric2F1[-3/2, 4, -1/2, 3/7 - (6*x)/7])/(201684*(1 - 2*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.31, size = 99, normalized size = 0.73 \begin {gather*} \frac {470205 (1-2 x)^5-4022865 (1-2 x)^4+12458691 (1-2 x)^3-15872031 (1-2 x)^2+5663616 (1-2 x)+1690304}{201684 (3 (1-2 x)-7)^4 (1-2 x)^{3/2}}-\frac {1935 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1690304 + 5663616*(1 - 2*x) - 15872031*(1 - 2*x)^2 + 12458691*(1 - 2*x)^3 - 4022865*(1 - 2*x)^4 + 470205*(1 -
2*x)^5)/(201684*(-7 + 3*(1 - 2*x))^4*(1 - 2*x)^(3/2)) - (1935*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/672
28

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fricas [A]  time = 1.25, size = 135, normalized size = 0.99 \begin {gather*} \frac {5805 \, \sqrt {7} \sqrt {3} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (1880820 \, x^{5} + 3343680 \, x^{4} + 1069281 \, x^{3} - 1034451 \, x^{2} - 611202 \, x - 48490\right )} \sqrt {-2 \, x + 1}}{2823576 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2823576*(5805*sqrt(7)*sqrt(3)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log((sqrt(7)*sqrt
(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(1880820*x^5 + 3343680*x^4 + 1069281*x^3 - 1034451*x^2 - 611202*x
- 48490)*sqrt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)

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giac [A]  time = 1.30, size = 121, normalized size = 0.89 \begin {gather*} \frac {1935}{941192} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (780 \, x - 467\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {3 \, {\left (141075 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 1076607 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 2765805 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2378705 \, \sqrt {-2 \, x + 1}\right )}}{7529536 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1935/941192*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/352947*(7
80*x - 467)/((2*x - 1)*sqrt(-2*x + 1)) - 3/7529536*(141075*(2*x - 1)^3*sqrt(-2*x + 1) + 1076607*(2*x - 1)^2*sq
rt(-2*x + 1) - 2765805*(-2*x + 1)^(3/2) + 2378705*sqrt(-2*x + 1))/(3*x + 2)^4

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maple [A]  time = 0.02, size = 84, normalized size = 0.62 \begin {gather*} -\frac {1935 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{470596}+\frac {176}{50421 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {2080}{117649 \sqrt {-2 x +1}}+\frac {\frac {423225 \left (-2 x +1\right )^{\frac {7}{2}}}{470596}-\frac {461403 \left (-2 x +1\right )^{\frac {5}{2}}}{67228}+\frac {169335 \left (-2 x +1\right )^{\frac {3}{2}}}{9604}-\frac {20805 \sqrt {-2 x +1}}{1372}}{\left (-6 x -4\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

176/50421/(-2*x+1)^(3/2)+2080/117649/(-2*x+1)^(1/2)+3888/117649*(5225/192*(-2*x+1)^(7/2)-119623/576*(-2*x+1)^(
5/2)+921935/1728*(-2*x+1)^(3/2)-2378705/5184*(-2*x+1)^(1/2))/(-6*x-4)^4-1935/470596*arctanh(1/7*21^(1/2)*(-2*x
+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.18, size = 128, normalized size = 0.94 \begin {gather*} \frac {1935}{941192} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {470205 \, {\left (2 \, x - 1\right )}^{5} + 4022865 \, {\left (2 \, x - 1\right )}^{4} + 12458691 \, {\left (2 \, x - 1\right )}^{3} + 15872031 \, {\left (2 \, x - 1\right )}^{2} + 11327232 \, x - 7353920}{201684 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 2401 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1935/941192*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/201684*(470205*(2*x
- 1)^5 + 4022865*(2*x - 1)^4 + 12458691*(2*x - 1)^3 + 15872031*(2*x - 1)^2 + 11327232*x - 7353920)/(81*(-2*x
+ 1)^(11/2) - 756*(-2*x + 1)^(9/2) + 2646*(-2*x + 1)^(7/2) - 4116*(-2*x + 1)^(5/2) + 2401*(-2*x + 1)^(3/2))

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mupad [B]  time = 1.23, size = 108, normalized size = 0.79 \begin {gather*} -\frac {1935\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{470596}-\frac {\frac {2752\,x}{3969}+\frac {1333\,{\left (2\,x-1\right )}^2}{1372}+\frac {3139\,{\left (2\,x-1\right )}^3}{4116}+\frac {2365\,{\left (2\,x-1\right )}^4}{9604}+\frac {1935\,{\left (2\,x-1\right )}^5}{67228}-\frac {5360}{11907}}{\frac {2401\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{7/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{9/2}}{3}+{\left (1-2\,x\right )}^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (1935*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/470596 - ((2752*x)/3969 + (1333*(2*x - 1)^2)/1372 + (313
9*(2*x - 1)^3)/4116 + (2365*(2*x - 1)^4)/9604 + (1935*(2*x - 1)^5)/67228 - 5360/11907)/((2401*(1 - 2*x)^(3/2))
/81 - (1372*(1 - 2*x)^(5/2))/27 + (98*(1 - 2*x)^(7/2))/3 - (28*(1 - 2*x)^(9/2))/3 + (1 - 2*x)^(11/2))

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

[Out]

Timed out

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