3.22.67 \(\int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\) [2167]

Optimal. Leaf size=140 \[ -\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}-\frac {177635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684 \sqrt {21}} \]

[Out]

11/21*(3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^4-177635/4235364*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/4116*(57
069+85754*x)/(2+3*x)^4/(1-2*x)^(1/2)-35527/12348*(1-2*x)^(1/2)/(2+3*x)^3-177635/172872*(1-2*x)^(1/2)/(2+3*x)^2
-177635/403368*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.03, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 149, 44, 65, 212} \begin {gather*} \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac {85754 x+57069}{4116 \sqrt {1-2 x} (3 x+2)^4}-\frac {177635 \sqrt {1-2 x}}{403368 (3 x+2)}-\frac {177635 \sqrt {1-2 x}}{172872 (3 x+2)^2}-\frac {35527 \sqrt {1-2 x}}{12348 (3 x+2)^3}-\frac {177635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-35527*Sqrt[1 - 2*x])/(12348*(2 + 3*x)^3) - (177635*Sqrt[1 - 2*x])/(172872*(2 + 3*x)^2) - (177635*Sqrt[1 - 2*
x])/(403368*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + (57069 + 85754*x)/(4116*Sqrt[1 -
2*x]*(2 + 3*x)^4) - (177635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(201684*Sqrt[21])

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 100

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

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :>
Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(
m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m +
1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] -
Dist[(a^2*d^2*f*h*(2 + 3*n + 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
*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b
*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h
}, x] && LtQ[m, -1] && LtQ[n, -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 {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {1}{21} \int \frac {(-167-315 x) (3+5 x)}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {35527}{588} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {177635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{12348}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {177635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{57624}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}+\frac {177635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{403368}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}-\frac {177635 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{403368}\\ &=-\frac {35527 \sqrt {1-2 x}}{12348 (2+3 x)^3}-\frac {177635 \sqrt {1-2 x}}{172872 (2+3 x)^2}-\frac {177635 \sqrt {1-2 x}}{403368 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac {57069+85754 x}{4116 \sqrt {1-2 x} (2+3 x)^4}-\frac {177635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{201684 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 75, normalized size = 0.54 \begin {gather*} \frac {-\frac {21 \left (-2094250-10307138 x-12952519 x^2+10906789 x^3+34105920 x^4+19184580 x^5\right )}{2 (1-2 x)^{3/2} (2+3 x)^4}-177635 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4235364} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-21*(-2094250 - 10307138*x - 12952519*x^2 + 10906789*x^3 + 34105920*x^4 + 19184580*x^5))/(2*(1 - 2*x)^(3/2)*
(2 + 3*x)^4) - 177635*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/4235364

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Maple [A]
time = 0.11, size = 84, normalized size = 0.60

method	result	size
risch	\(\frac {19184580 x^{5}[phone] x^{4}[phone] x^{3}-12952519 x^{2}-10307138 x -2094250}{403368 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {177635 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4235364}\)	\(68\)
derivativedivides	\(\frac {5324}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {29040}{117649 \sqrt {1-2 x}}+\frac {\frac {1782045 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1707607 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {1636345 \left (1-2 x \right )^{\frac {3}{2}}}{28812}-\frac {174235 \sqrt {1-2 x}}{4116}}{\left (-4-6 x \right )^{4}}-\frac {177635 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4235364}\)	\(84\)
default	\(\frac {5324}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {29040}{117649 \sqrt {1-2 x}}+\frac {\frac {1782045 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1707607 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {1636345 \left (1-2 x \right )^{\frac {3}{2}}}{28812}-\frac {174235 \sqrt {1-2 x}}{4116}}{\left (-4-6 x \right )^{4}}-\frac {177635 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4235364}\)	\(84\)
trager	\(-\frac {\left (19184580 x^{5}[phone] x^{4}[phone] x^{3}-12952519 x^{2}-10307138 x -2094250\right ) \sqrt {1-2 x}}{403368 \left (2+3 x \right )^{4} \left (-1+2 x \right )^{2}}-\frac {177635 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{8470728}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5324/50421/(1-2*x)^(3/2)+29040/117649/(1-2*x)^(1/2)+324/117649*(198005/144*(1-2*x)^(7/2)-11953249/1296*(1-2*x)
^(5/2)[phone]/3888*(1-2*x)^(3/2)-59762605/3888*(1-2*x)^(1/2))/(-4-6*x)^4-177635/4235364*arctanh(1/7*21^(1/2)
*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.49, size = 128, normalized size = 0.91 \begin {gather*} \frac {177635}{8470728} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4796145 \, {\left (2 \, x - 1\right )}^{5} + 41033685 \, {\left (2 \, x - 1\right )}^{4} + 127080079 \, {\left (2 \, x - 1\right )}^{3} + 157094539 \, {\left (2 \, x - 1\right )}^{2} + 63748608 \, x - 83006000}{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)^3/(1-2*x)^(5/2)/(2+3*x)^5,x, algorithm="maxima")

[Out]

177635/8470728*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/201684*(4796145*
(2*x - 1)^5 + 41033685*(2*x - 1)^4 + 127080079*(2*x - 1)^3 + 157094539*(2*x - 1)^2 + 63748608*x - 83006000)/(8
1*(-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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Fricas [A]
time = 1.50, size = 129, normalized size = 0.92 \begin {gather*} \frac {177635 \, \sqrt {21} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (19184580 \, x^{5} + 34105920 \, x^{4} + 10906789 \, x^{3} - 12952519 \, x^{2} - 10307138 \, x - 2094250\right )} \sqrt {-2 \, x + 1}}{8470728 \, {\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)^3/(1-2*x)^(5/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

1/8470728*(177635*sqrt(21)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log((3*x + sqrt(21)*sq
rt(-2*x + 1) - 5)/(3*x + 2)) - 21*(19184580*x^5 + 34105920*x^4 + 10906789*x^3 - 12952519*x^2 - 10307138*x - 20
94250)*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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Sympy [F(-1)] Timed out
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)**3/(1-2*x)**(5/2)/(2+3*x)**5,x)

[Out]

Timed out

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Giac [A]
time = 4.00, size = 121, normalized size = 0.86 \begin {gather*} \frac {177635}{8470728} \, \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 {484 \, {\left (360 \, x - 257\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {5346135 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 35859747 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 80180905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 59762605 \, \sqrt {-2 \, x + 1}}{22588608 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

177635/8470728*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 484/35294
7*(360*x - 257)/((2*x - 1)*sqrt(-2*x + 1)) - 1/22588608*(5346135*(2*x - 1)^3*sqrt(-2*x + 1) + 35859747*(2*x -
1)^2*sqrt(-2*x + 1) - 80180905*(-2*x + 1)^(3/2) + 59762605*sqrt(-2*x + 1))/(3*x + 2)^4

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Mupad [B]
time = 1.24, size = 108, normalized size = 0.77 \begin {gather*} -\frac {177635\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4235364}-\frac {\frac {15488\,x}{3969}+\frac {3206011\,{\left (2\,x-1\right )}^2}{333396}+\frac {2593471\,{\left (2\,x-1\right )}^3}{333396}+\frac {1953985\,{\left (2\,x-1\right )}^4}{777924}+\frac {177635\,{\left (2\,x-1\right )}^5}{605052}-\frac {60500}{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)^3/((1 - 2*x)^(5/2)*(3*x + 2)^5),x)

[Out]

- (177635*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4235364 - ((15488*x)/3969 + (3206011*(2*x - 1)^2)/3333
96 + (2593471*(2*x - 1)^3)/333396 + (1953985*(2*x - 1)^4)/777924 + (177635*(2*x - 1)^5)/605052 - 60500/11907)/
((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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