3.67 \(\int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3} \, dx\)
Optimal. Leaf size=225 \[ -\frac{24007 \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{6608797 \sqrt{66} \sqrt{2 x-5}}-\frac{223825 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1030972332 (5 x+7)}-\frac{25 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{55614 (5 x+7)^2}+\frac{44765 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{515486166 \sqrt{5-2 x}}-\frac{48493305 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{21306761528 \sqrt{11} \sqrt{2 x-5}} \]
[Out]
(-25*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(55614*(7 + 5*x)^2) - (223825*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sq
rt[1 + 4*x])/(1030972332*(7 + 5*x)) + (44765*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[1
1]], -1/2])/(515486166*Sqrt[5 - 2*x]) - (24007*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])
/(6608797*Sqrt[66]*Sqrt[-5 + 2*x]) - (48493305*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[
11]], -1/2])/(21306761528*Sqrt[11]*Sqrt[-5 + 2*x])
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Rubi [A] time = 0.309706, antiderivative size = 225, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used
= {172, 1604, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ -\frac{223825 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1030972332 (5 x+7)}-\frac{25 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{55614 (5 x+7)^2}-\frac{24007 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{6608797 \sqrt{66} \sqrt{2 x-5}}+\frac{44765 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{515486166 \sqrt{5-2 x}}-\frac{48493305 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{21306761528 \sqrt{11} \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3),x]
[Out]
(-25*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(55614*(7 + 5*x)^2) - (223825*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sq
rt[1 + 4*x])/(1030972332*(7 + 5*x)) + (44765*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[1
1]], -1/2])/(515486166*Sqrt[5 - 2*x]) - (24007*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])
/(6608797*Sqrt[66]*Sqrt[-5 + 2*x]) - (48493305*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[
11]], -1/2])/(21306761528*Sqrt[11]*Sqrt[-5 + 2*x])
Rule 172
Int[((a_.) + (b_.)*(x_))^(m_)/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_
Symbol] :> Simp[(b^2*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e -
a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c
+ d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(
2*m + 3)*(d*e*g + c*f*g + c*e*h) - 2*b*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h))*x + d*f*h*(2*m +
5)*b^2*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[2*m] && LeQ[m, -2]
Rule 1604
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_
.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]
*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d
)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*
d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - (b*B - a*C)*(
a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h +
c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h
*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[
2*m] && LtQ[m, -1]
Rule 1607
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h
*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q,
x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]
Rule 168
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]
Rule 538
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] && !GtQ[c, 0]
Rule 537
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])
Rule 158
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]
Rule 114
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) && !LtQ[-((b*c - a*d)/d), 0]
Rule 113
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !LtQ[-((b*c - a*d)/d),
0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)
/b, 0])
Rule 121
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]
Rule 119
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) && !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^3} \, dx &=-\frac{25 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{55614 (7+5 x)^2}+\frac{\int \frac{16079-6860 x+600 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^2} \, dx}{111228}\\ &=-\frac{25 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{55614 (7+5 x)^2}-\frac{223825 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1030972332 (7+5 x)}+\frac{\int \frac{123236883-38449440 x-16115400 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx}{6185833992}\\ &=-\frac{25 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{55614 (7+5 x)^2}-\frac{223825 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1030972332 (7+5 x)}+\frac{\int \frac{-3177576-3223080 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{6185833992}+\frac{16164435 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx}{687314888}\\ &=-\frac{25 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{55614 (7+5 x)^2}-\frac{223825 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1030972332 (7+5 x)}-\frac{44765 \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx}{171828722}-\frac{24007 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{13217594}-\frac{16164435 \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{-\frac{11}{3}-\frac{2 x^2}{3}}} \, dx,x,\sqrt{2-3 x}\right )}{343657444}\\ &=-\frac{25 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{55614 (7+5 x)^2}-\frac{223825 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1030972332 (7+5 x)}-\frac{\left (16164435 \sqrt{\frac{3}{11}} \sqrt{5-2 x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{1+\frac{2 x^2}{11}}} \, dx,x,\sqrt{2-3 x}\right )}{343657444 \sqrt{-5+2 x}}-\frac{\left (24007 \sqrt{5-2 x}\right ) \int \frac{1}{\sqrt{2-3 x} \sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{6608797 \sqrt{22} \sqrt{-5+2 x}}-\frac{\left (44765 \sqrt{-5+2 x}\right ) \int \frac{\sqrt{\frac{15}{11}-\frac{6 x}{11}}}{\sqrt{2-3 x} \sqrt{\frac{3}{11}+\frac{12 x}{11}}} \, dx}{171828722 \sqrt{5-2 x}}\\ &=-\frac{25 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{55614 (7+5 x)^2}-\frac{223825 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{1030972332 (7+5 x)}+\frac{44765 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{515486166 \sqrt{5-2 x}}-\frac{24007 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{6608797 \sqrt{66} \sqrt{-5+2 x}}-\frac{48493305 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{21306761528 \sqrt{11} \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.438753, size = 147, normalized size = 0.65 \[ \frac{-\sqrt{55-22 x} (5 x+7)^2 \left (61059460 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )-3 \left (38699284 \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right ),-\frac{1}{2}\right )+48493305 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )\right )-17050 \sqrt{2-3 x} (2 x-5) \sqrt{4 x+1} (44765 x+81209)}{703123130424 \sqrt{2 x-5} (5 x+7)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^3),x]
[Out]
(-17050*Sqrt[2 - 3*x]*(-5 + 2*x)*Sqrt[1 + 4*x]*(81209 + 44765*x) - Sqrt[55 - 22*x]*(7 + 5*x)^2*(61059460*Ellip
ticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 3*(38699284*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/
2] + 48493305*EllipticPi[55/124, -ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])))/(703123130424*Sqrt[-5 + 2*x]*(7
+ 5*x)^2)
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Maple [B] time = 0.02, size = 461, normalized size = 2.1 \begin{align*} -{\frac{1}{ \left ( 16874955130176\,{x}^{3}-49218619129680\,{x}^{2}+14765585738904\,x+7031231304240 \right ) \left ( 7+5\,x \right ) ^{2}}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 2902446300\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ){x}^{2}-1526486500\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ){x}^{2}-3636997875\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ){x}^{2}+8126849640\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) x-4274162200\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) x-10183594050\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ) x+5688794748\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -2991913540\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -7128515835\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ) +18317838000\,{x}^{4}-20196304700\,{x}^{3}-80894833250\,{x}^{2}+36709314950\,x+13846134500 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(7+5*x)^3/(2-3*x)^(1/2)/(2*x-5)^(1/2)/(4*x+1)^(1/2),x)
[Out]
-1/703123130424*(2-3*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(2902446300*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*
x+1)^(1/2)*EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x^2-1526486500*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(
4*x+1)^(1/2)*EllipticE(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x^2-3636997875*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)
*(4*x+1)^(1/2)*EllipticPi(2/11*(22-33*x)^(1/2),55/124,1/2*I*2^(1/2))*x^2+8126849640*11^(1/2)*(2-3*x)^(1/2)*(5-
2*x)^(1/2)*(4*x+1)^(1/2)*EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x-4274162200*11^(1/2)*(2-3*x)^(1/2)*(5-
2*x)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))*x-10183594050*11^(1/2)*(2-3*x)^(1/2)*(5
-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(22-33*x)^(1/2),55/124,1/2*I*2^(1/2))*x+5688794748*11^(1/2)*(2-3*x)^
(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))-2991913540*11^(1/2)*(2-3*x)^(1
/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticE(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))-7128515835*11^(1/2)*(2-3*x)^(1/2
)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*(22-33*x)^(1/2),55/124,1/2*I*2^(1/2))+18317838000*x^4-2019630470
0*x^3-80894833250*x^2+36709314950*x+13846134500)/(24*x^3-70*x^2+21*x+10)/(7+5*x)^2
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(7+5*x)^3/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((5*x + 7)^3*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{3000 \, x^{6} + 3850 \, x^{5} - 16485 \, x^{4} - 30943 \, x^{3} - 3325 \, x^{2} + 14553 \, x + 3430}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(7+5*x)^3/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="fricas")
[Out]
integral(-sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(3000*x^6 + 3850*x^5 - 16485*x^4 - 30943*x^3 - 3325*x^2 +
14553*x + 3430), x)
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(7+5*x)**3/(2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)
[Out]
Exception raised: ValueError
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 7\right )}^{3} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(7+5*x)^3/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="giac")
[Out]
integrate(1/((5*x + 7)^3*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), x)