3.19.22 \(\int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx\) [1822]

Optimal. Leaf size=100 \[ -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]

[Out]

7559/11907*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-53/63*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)-1/6*(3+5*x)^3*(1
-2*x)^(1/2)/(2+3*x)^2+5/1134*(323+2815*x)*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 100, 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 = {99, 154, 152, 65, 212} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^3}{6 (3 x+2)^2}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)}+\frac {5 \sqrt {1-2 x} (2815 x+323)}{1134}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(6*(2 + 3*x)^2) + (5*Sqrt[1 - 2*x
]*(323 + 2815*x))/1134 + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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 99

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

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

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)^3}{(2+3 x)^3} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {1}{126} \int \frac {(643-2815 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}-\frac {7559 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1134}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}+\frac {7559 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1134}\\ &=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (323+2815 x)}{1134}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 63, normalized size = 0.63 \begin {gather*} \frac {\sqrt {1-2 x} \left (-15815-32833 x+7350 x^2+31500 x^3\right )}{1134 (2+3 x)^2}+\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-15815 - 32833*x + 7350*x^2 + 31500*x^3))/(1134*(2 + 3*x)^2) + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(567*Sqrt[21])

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Maple [A]
time = 0.12, size = 66, normalized size = 0.66

method	result	size
risch	\(-\frac {63000 x^{4}-16800 x^{3}-73016 x^{2}+1203 x +15815}{1134 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {7559 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\)	\(56\)
derivativedivides	\(-\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {50 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {211 \left (1-2 x \right )^{\frac {3}{2}}}{126}+\frac {209 \sqrt {1-2 x}}{54}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {7559 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\)	\(66\)
default	\(-\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {50 \sqrt {1-2 x}}{27}-\frac {2 \left (-\frac {211 \left (1-2 x \right )^{\frac {3}{2}}}{126}+\frac {209 \sqrt {1-2 x}}{54}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {7559 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\)	\(66\)
trager	\(\frac {\left (31500 x^{3}+7350 x^{2}-32833 x -15815\right ) \sqrt {1-2 x}}{1134 \left (2+3 x \right )^{2}}+\frac {7559 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{23814}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/81*(1-2*x)^(3/2)-50/27*(1-2*x)^(1/2)-2/3*(-211/126*(1-2*x)^(3/2)+209/54*(1-2*x)^(1/2))/(-4-6*x)^2+7559/11
907*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.52, size = 92, normalized size = 0.92 \begin {gather*} -\frac {125}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7559}{23814} \, \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 {633 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{567 \, {\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)^3*(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="maxima")

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 50/27*sqrt(-2*x + 1) + 1/567*(633*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]
time = 0.67, size = 80, normalized size = 0.80 \begin {gather*} \frac {7559 \, \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 (31500 \, x^{3} + 7350 \, x^{2} - 32833 \, x - 15815\right )} \sqrt {-2 \, x + 1}}{23814 \, {\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)^3*(1-2*x)^(1/2)/(2+3*x)^3,x, algorithm="fricas")

[Out]

1/23814*(7559*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(31500*x^3 +
7350*x^2 - 32833*x - 15815)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [A]
time = 224.53, size = 374, normalized size = 3.74 \begin {gather*} - \frac {125 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {50 \sqrt {1 - 2 x}}{27} - \frac {428 \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 )}{81} - \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 )}{81} - \frac {370 \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 )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*(1 - 2*x)**(3/2)/81 - 50*sqrt(1 - 2*x)/27 - 428*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)))/81 - 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)))/81 - 370*Piecewise(
(-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, x < -2/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, x > -
2/3))/27

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Giac [A]
time = 1.06, size = 86, normalized size = 0.86 \begin {gather*} -\frac {125}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7559}{23814} \, \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 {633 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{2268 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*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) + 1/2268*(633*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.07, size = 74, normalized size = 0.74 \begin {gather*} -\frac {50\,\sqrt {1-2\,x}}{27}-\frac {125\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {\frac {209\,\sqrt {1-2\,x}}{729}-\frac {211\,{\left (1-2\,x\right )}^{3/2}}{1701}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,7559{}\mathrm {i}}{11907} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*7559i)/11907 - (50*(1 - 2*x)^(1/2))/27 - (125*(1 - 2*x)^(3/2
))/81 - ((209*(1 - 2*x)^(1/2))/729 - (211*(1 - 2*x)^(3/2))/1701)/((28*x)/3 + (2*x - 1)^2 + 7/9)

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