3.19.10 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx\) [1810]

Optimal. Leaf size=81 \[ \frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \]

[Out]

-1/126*(1-2*x)^(3/2)/(2+3*x)^2+139/882*(1-2*x)^(3/2)/(2+3*x)-863/1323*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(
1/2)+863/441*(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 52, 65, 212} \begin {gather*} \frac {139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac {863}{441} \sqrt {1-2 x}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(863*Sqrt[1 - 2*x])/441 - (1 - 2*x)^(3/2)/(126*(2 + 3*x)^2) + (139*(1 - 2*x)^(3/2))/(882*(2 + 3*x)) - (863*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt[21])

Rule 52

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 65

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 79

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] || L
tQ[p, n]))))

Rule 91

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 212

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {\sqrt {1-2 x} (561+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac {863}{294} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac {863}{126} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863}{126} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 58, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1-2 x} \left (1025+2941 x+2100 x^2\right )}{126 (2+3 x)^2}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(1025 + 2941*x + 2100*x^2))/(126*(2 + 3*x)^2) - (863*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(63*Sqrt
[21])

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Maple [A]
time = 0.11, size = 57, normalized size = 0.70

method	result	size
risch	\(-\frac {4200 x^{3}+3782 x^{2}-891 x -1025}{126 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {863 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}\)	\(51\)
derivativedivides	\(\frac {50 \sqrt {1-2 x}}{27}+\frac {-\frac {47 \left (1-2 x \right )^{\frac {3}{2}}}{21}+\frac {139 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{2}}-\frac {863 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}\)	\(57\)
default	\(\frac {50 \sqrt {1-2 x}}{27}+\frac {-\frac {47 \left (1-2 x \right )^{\frac {3}{2}}}{21}+\frac {139 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{2}}-\frac {863 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1323}\)	\(57\)
trager	\(\frac {\left (2100 x^{2}+2941 x +1025\right ) \sqrt {1-2 x}}{126 \left (2+3 x \right )^{2}}+\frac {863 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2646}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/27*(1-2*x)^(1/2)+2/3*(-47/14*(1-2*x)^(3/2)+139/18*(1-2*x)^(1/2))/(-4-6*x)^2-863/1323*arctanh(1/7*21^(1/2)*(
1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.64, size = 83, normalized size = 1.02 \begin {gather*} \frac {863}{2646} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{189 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

863/2646*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 50/27*sqrt(-2*x + 1) - 1
/189*(423*(-2*x + 1)^(3/2) - 973*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]
time = 1.45, size = 74, normalized size = 0.91 \begin {gather*} \frac {863 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (2100 \, x^{2} + 2941 \, x + 1025\right )} \sqrt {-2 \, x + 1}}{2646 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2646*(863*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(2100*x^2 + 29
41*x + 1025)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [A]
time = 178.23, size = 360, normalized size = 4.44 \begin {gather*} \frac {50 \sqrt {1 - 2 x}}{27} + \frac {32 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} + \frac {56 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{27} + \frac {130 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50*sqrt(1 - 2*x)/27 + 32*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x
)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x)
> -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3 + 56*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1
)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1
- 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (
sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/27 + 130*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqr
t(1 - 2*x)/7)/21, x < -2/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, x > -2/3))/9

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Giac [A]
time = 0.57, size = 77, normalized size = 0.95 \begin {gather*} \frac {863}{2646} \, \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 {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{756 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

863/2646*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 50/27*sqrt(-2*x
+ 1) - 1/756*(423*(-2*x + 1)^(3/2) - 973*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.06, size = 62, normalized size = 0.77 \begin {gather*} \frac {50\,\sqrt {1-2\,x}}{27}-\frac {863\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1323}+\frac {\frac {139\,\sqrt {1-2\,x}}{243}-\frac {47\,{\left (1-2\,x\right )}^{3/2}}{189}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(50*(1 - 2*x)^(1/2))/27 - (863*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1323 + ((139*(1 - 2*x)^(1/2))/243
- (47*(1 - 2*x)^(3/2))/189)/((28*x)/3 + (2*x - 1)^2 + 7/9)

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