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

Optimal. Leaf size=330 \[ -\frac{2852696 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{11598438735 \sqrt{2 x-5}}+\frac{1426348 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{2319687747 \sqrt{5 x+7}}+\frac{17906 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{2085525 (5 x+7)^{3/2}}-\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{25 (5 x+7)^{5/2}}-\frac{48884 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{9593415 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{1426348 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{297395865 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]

[Out]

(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(25*(7 + 5*x)^(5/2)) + (17906*Sq
rt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(2085525*(7 + 5*x)^(3/2)) + (1426348*S
qrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(2319687747*Sqrt[7 + 5*x]) - (2852696
*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(11598438735*Sqrt[-5 + 2*x]) + (1426
348*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[3
9/23]*Sqrt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(297395865*Sqrt[(2 - 3*x)/(5 - 2*
x)]*Sqrt[7 + 5*x]) - (48884*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 +
4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(9593415*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(
5 - 2*x)])

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Rubi [A]  time = 1.0317, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216 \[ -\frac{2852696 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{11598438735 \sqrt{2 x-5}}+\frac{1426348 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{2319687747 \sqrt{5 x+7}}+\frac{17906 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{2085525 (5 x+7)^{3/2}}-\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{25 (5 x+7)^{5/2}}-\frac{48884 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{9593415 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{1426348 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{297395865 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(25*(7 + 5*x)^(5/2)) + (17906*Sq
rt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(2085525*(7 + 5*x)^(3/2)) + (1426348*S
qrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(2319687747*Sqrt[7 + 5*x]) - (2852696
*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(11598438735*Sqrt[-5 + 2*x]) + (1426
348*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[3
9/23]*Sqrt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(297395865*Sqrt[(2 - 3*x)/(5 - 2*
x)]*Sqrt[7 + 5*x]) - (48884*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 +
4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(9593415*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(
5 - 2*x)])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 2.05532, size = 251, normalized size = 0.76 \[ -\frac{2 \sqrt{2 x-5} \sqrt{4 x+1} \left (-236555 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^3 F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+713174 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^3 E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+31 \sqrt{\frac{5 x+7}{3 x-2}} \left (50105384 x^4-729949210 x^3+1137407943 x^2+880765228 x+137502130\right )\right )}{11598438735 \sqrt{2-3 x} (5 x+7)^{5/2} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(31*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(137502130 + 880
765228*x + 1137407943*x^2 - 729949210*x^3 + 50105384*x^4) + 713174*Sqrt[682]*(-2
+ 3*x)*(7 + 5*x)^3*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[ArcSin[Sqrt[
31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] - 236555*Sqrt[682]*(-2 + 3*x)*(7 + 5
*x)^3*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticF[ArcSin[Sqrt[31/39]*Sqrt[(-
5 + 2*x)/(-2 + 3*x)]], 39/62]))/(11598438735*Sqrt[2 - 3*x]*(7 + 5*x)^(5/2)*Sqrt[
(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2))

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Maple [B]  time = 0.059, size = 1033, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/11598438735*(17811200*((-5+2*x)/(1+4*x))^(1/2)*3^(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))*13^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*x^4+285269600*((-5
+2*x)/(1+4*x))^(1/2)*3^(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))*13^(1/2)*
11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*x^4+58776960*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))*11^(1/2)*((7+5*x)/(1+4*x))^(1/
2)*x^3+941389680*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))*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*x^3+60958832*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))[phone]*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*1
3^(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))+2
0571936*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))[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*E
llipticE(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))+2181872*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))[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)*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))-3514495404*x^4+19294337060*x^3-26198770563*x^2-385527
4122*x+9191461480)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(2-3*x)^(1/2)/(120*x^4-182*x^3-3
85*x^2+197*x+70)/(7+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343\right )} \sqrt{5 \, x + 7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/((125*x^3 + 525*x^2 + 735*x
+ 343)*sqrt(5*x + 7)), 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((2-3*x)**(1/2)*(-5+2*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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