3.27.35 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\) [2635]

Optimal. Leaf size=152 \[ -\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {985525 \sqrt {1-2 x}}{1369599 (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {95783075 \sqrt {1-2 x}}{15065589 \sqrt {3+5 x}}-\frac {14985 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \]

[Out]

-190/1617/(1-2*x)^(3/2)/(3+5*x)^(3/2)+3/7/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^(3/2)-14985/2401*arctan(1/7*(1-2*x)^(1
/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1090/41503/(3+5*x)^(3/2)/(1-2*x)^(1/2)-985525/1369599*(1-2*x)^(1/2)/(3+5*x)
^(3/2)[phone]/15065589*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {105, 157, 12, 95, 210} \begin {gather*} -\frac {14985 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}}+\frac {95783075 \sqrt {1-2 x}}{15065589 \sqrt {5 x+3}}-\frac {985525 \sqrt {1-2 x}}{1369599 (5 x+3)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac {190}{1617 (1-2 x)^{3/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-190/(1617*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 1090/(41503*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (985525*Sqrt[1 - 2*
x])/(1369599*(3 + 5*x)^(3/2)) + 3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2)) + (95783075*Sqrt[1 - 2*x])/(15
065589*Sqrt[3 + 5*x]) - (14985*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

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

Rule 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 210

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

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx &=\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {1}{7} \int \frac {\frac {25}{2}-120 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac {2 \int \frac {-\frac {6915}{4}+4275 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx}{1617}\\ &=-\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {4 \int \frac {\frac {473595}{8}-24525 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{124509}\\ &=-\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {985525 \sqrt {1-2 x}}{1369599 (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {36182505}{16}-\frac {8869725 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{4108797}\\ &=-\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {985525 \sqrt {1-2 x}}{1369599 (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {95783075 \sqrt {1-2 x}}{15065589 \sqrt {3+5 x}}+\frac {16 \int \frac {1974558465}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{45196767}\\ &=-\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {985525 \sqrt {1-2 x}}{1369599 (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {95783075 \sqrt {1-2 x}}{15065589 \sqrt {3+5 x}}+\frac {14985}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {985525 \sqrt {1-2 x}}{1369599 (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {95783075 \sqrt {1-2 x}}{15065589 \sqrt {3+5 x}}+\frac {14985}{343} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {190}{1617 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1090}{41503 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {985525 \sqrt {1-2 x}}{1369599 (3+5 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {95783075 \sqrt {1-2 x}}{15065589 \sqrt {3+5 x}}-\frac {14985 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 84, normalized size = 0.55 \begin {gather*} \frac {555141781-429626520 x-3498236655 x^2+1402439900 x^3+5746984500 x^4}{15065589 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac {14985 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(555141781 - 429626520*x - 3498236655*x^2 + 1402439900*x^3 + 5746984500*x^4)/(15065589*(1 - 2*x)^(3/2)*(2 + 3*
x)*(3 + 5*x)^(3/2)) - (14985*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(113)=226\).
time = 0.09, size = 305, normalized size = 2.01

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \left (197455846500 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-90171503235 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}-89513317080 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -48975313170 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-6014771280 x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{210918246 \left (2+3 x \right ) \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210918246*(1-2*x)^(1/2)*(197455846500*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+1711284
00300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-90171503235*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+80457783000*x^4*(-10*x^2-x+3)^(1/2)-89513317080*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+19634158600*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-48975313170*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))-6014771280*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(2+3*x)/(-1+2*x
)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]
time = 0.51, size = 121, normalized size = 0.80 \begin {gather*} \frac {14985}{4802} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {191566150 \, x}{15065589 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {100119385}{15065589 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {57250 \, x}{17787 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {3}{7 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {30715}{17787 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14985/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 191566150/15065589*x/sqrt(-10*x^2 - x +
3) + 100119385/15065589/sqrt(-10*x^2 - x + 3) + 57250/17787*x/(-10*x^2 - x + 3)^(3/2) + 3/7/(3*(-10*x^2 - x +
3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 30715/17787/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]
time = 0.52, size = 131, normalized size = 0.86 \begin {gather*} -\frac {658186155 \, \sqrt {7} {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (5746984500 \, x^{4} + 1402439900 \, x^{3} - 3498236655 \, x^{2} - 429626520 \, x + 555141781\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{210918246 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210918246*(658186155*sqrt(7)*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*arctan(1/14*sqrt(7)*(37*x
+ 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(5746984500*x^4 + 1402439900*x^3 - 3498236655*x^2 -
429626520*x + 555141781)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (113) = 226\).
time = 1.40, size = 348, normalized size = 2.29 \begin {gather*} \frac {2997}{9604} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {125}{702768} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {1440 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {5760 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {5346 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{343 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} - \frac {32 \, {\left (956 \, \sqrt {5} {\left (5 \, x + 3\right )} - 5643 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{376639725 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2997/9604*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 125/702768*sqrt(10)*(((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 1440*(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22))/sqrt(5*x + 3) + 5760*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 5346/343*sqrt(10)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 28
0) - 32/376639725*(956*sqrt(5)*(5*x + 3) - 5643*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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