3.85 \(\int \frac{\sqrt{2-3 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{\sqrt{-5+2 x}} \, dx\)

Optimal. Leaf size=429 \[ \frac{1}{8} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{5/2}+\frac{1445}{576} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{3/2}+\frac{1561915 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}}{27648}+\frac{2466927 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{4096 \sqrt{2 x-5}}+\frac{861015607 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{331776 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{2466927 \sqrt{429} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{8192 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{331574321009 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{1658880 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

[Out]

(2466927*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(4096*Sqrt[-5 + 2*x]) + (156
1915*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/27648 + (1445*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(3/2))/576 + (Sqrt[2 - 3*x]*Sq
rt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/8 - (2466927*Sqrt[429]*Sqrt[2 - 3*x]
*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt[-5
+ 2*x]], -23/39])/(8192*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (861015607*Sq
rt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])],
-39/23])/(331776*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (331574321009*(2 -
3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55,
ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(1658880*Sqrt[429]*S
qrt[-5 + 2*x]*Sqrt[1 + 4*x])

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Rubi [A]  time = 1.49438, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.297 \[ \frac{1}{8} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{5/2}+\frac{1445}{576} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{3/2}+\frac{1561915 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}}{27648}+\frac{2466927 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{4096 \sqrt{2 x-5}}+\frac{861015607 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{331776 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{2466927 \sqrt{429} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{8192 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{331574321009 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{1658880 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/Sqrt[-5 + 2*x],x]

[Out]

(2466927*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(4096*Sqrt[-5 + 2*x]) + (156
1915*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/27648 + (1445*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(3/2))/576 + (Sqrt[2 - 3*x]*Sq
rt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/8 - (2466927*Sqrt[429]*Sqrt[2 - 3*x]
*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt[-5
+ 2*x]], -23/39])/(8192*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (861015607*Sq
rt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])],
-39/23])/(331776*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (331574321009*(2 -
3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55,
ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(1658880*Sqrt[429]*S
qrt[-5 + 2*x]*Sqrt[1 + 4*x])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 3 x + 2} \sqrt{4 x + 1} \left (5 x + 7\right )^{\frac{5}{2}}}{\sqrt{2 x - 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((7+5*x)**(5/2)*(2-3*x)**(1/2)*(1+4*x)**(1/2)/(-5+2*x)**(1/2),x)

[Out]

Integral(sqrt(-3*x + 2)*sqrt(4*x + 1)*(5*x + 7)**(5/2)/sqrt(2*x - 5), x)

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Mathematica [A]  time = 3.75511, size = 345, normalized size = 0.8 \[ -\frac{\sqrt{2 x-5} \sqrt{4 x+1} \left (10666876180 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-12388907394 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+\sqrt{\frac{5 x+7}{3 x-2}} \left (10695945839 \sqrt{682} \sqrt{\frac{4 x+1}{3 x-2}} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} (2-3 x)^2 \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+186 \left (82944000 x^6+439372800 x^5+1004819520 x^4+3088122056 x^3-12645389558 x^2-26349657233 x-5752341805\right )\right )\right )}{41140224 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/Sqrt[-5 + 2*x],x]

[Out]

-(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(-12388907394*Sqrt[682]*Sqrt[(-5 - 18*x + 8*x^2)/
(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/
(-2 + 3*x)]], 39/62] + 10666876180*Sqrt[682]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^
2]*(-14 + 11*x + 15*x^2)*EllipticF[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)
]], 39/62] + Sqrt[(7 + 5*x)/(-2 + 3*x)]*(186*(-5752341805 - 26349657233*x - 1264
5389558*x^2 + 3088122056*x^3 + 1004819520*x^4 + 439372800*x^5 + 82944000*x^6) +
10695945839*Sqrt[682]*(2 - 3*x)^2*Sqrt[(1 + 4*x)/(-2 + 3*x)]*Sqrt[(-35 - 11*x +
10*x^2)/(2 - 3*x)^2]*EllipticPi[117/62, ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 +
3*x)]], 39/62])))/(41140224*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]*Sqrt[(7 + 5*x)/(-2 + 3*
x)]*(-5 - 18*x + 8*x^2))

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Maple [B]  time = 0.057, size = 944, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((7+5*x)^(5/2)*(2-3*x)^(1/2)*(1+4*x)^(1/2)/(-5+2*x)^(1/2),x)

[Out]

-1/948879360*(7+5*x)^(1/2)*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(424170712
240*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*(
(-2+3*x)/(1+4*x))^(1/2)*x^2*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(
1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-3936108068752*11^(1/2)*((7+5*x)/(1+
4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*x
^2*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39*2^(1/2)
*3^(1/2)*31^(1/2)*13^(1/2))-4571906470560*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/
2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*x^2*EllipticE(1/31
*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2
))[phone]*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4
*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*x*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/
(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-1968054034376*11^(1/2)*((
7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x
))^(1/2)*x*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39
*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-2285953235280*11^(1/2)*((7+5*x)/(1+4*x))^(1/
2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*x*Elliptic
E(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*1
3^(1/2))[phone]*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)
/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*
x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-246006754297*11^(1/2)*
((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4
*x))^(1/2)*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39
*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-285744154410*11^(1/2)*((7+5*x)/(1+4*x))^(1/2
)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*EllipticE(1
/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(
1/2))-355829760000*x^6-1884909312000*x^5-4310675740800*x^4-13248043620240*x^3+85
680578188920*x^2+78464986845960*x-85333953104400)/(120*x^4-182*x^3-385*x^2+197*x
+70)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 7\right )}^{\frac{5}{2}} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 7)^(5/2)*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5),x, algorithm="maxima")

[Out]

integrate((5*x + 7)^(5/2)*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 7)^(5/2)*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5),x, algorithm="fricas")

[Out]

integral((25*x^2 + 70*x + 49)*sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*
x - 5), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((7+5*x)**(5/2)*(2-3*x)**(1/2)*(1+4*x)**(1/2)/(-5+2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 7\right )}^{\frac{5}{2}} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 7)^(5/2)*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5),x, algorithm="giac")

[Out]

integrate((5*x + 7)^(5/2)*sqrt(4*x + 1)*sqrt(-3*x + 2)/sqrt(2*x - 5), x)