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

Optimal. Leaf size=330 \[ \frac{8185936 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{90467822133 \sqrt{2 x-5}}-\frac{20464840 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{90467822133 \sqrt{5 x+7}}-\frac{3646 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{16267095 (5 x+7)^{3/2}}+\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{195 (5 x+7)^{5/2}}+\frac{111628 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{74828637 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{4092968 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{2319687747 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]

[Out]

(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(195*(7 + 5*x)^(5/2)) - (3646*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(16267095*(7 + 5*x)^(3/2)) - (20464840*
Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(90467822133*Sqrt[7 + 5*x]) + (81859
36*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(90467822133*Sqrt[-5 + 2*x]) - (40
92968*Sqrt[11/39]*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])/(2319687747*Sqrt[(2 - 3*x)/(5 -
2*x)]*Sqrt[7 + 5*x]) + (111628*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[
1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(74828637*Sqrt[-5 + 2*x]*Sqrt[(7 + 5
*x)/(5 - 2*x)])

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Rubi [A]  time = 1.04261, 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 \[ \frac{8185936 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{90467822133 \sqrt{2 x-5}}-\frac{20464840 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{90467822133 \sqrt{5 x+7}}-\frac{3646 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{16267095 (5 x+7)^{3/2}}+\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{195 (5 x+7)^{5/2}}+\frac{111628 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{74828637 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{4092968 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{2319687747 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(195*(7 + 5*x)^(5/2)) - (3646*Sqr
t[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(16267095*(7 + 5*x)^(3/2)) - (20464840*
Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(90467822133*Sqrt[7 + 5*x]) + (81859
36*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(90467822133*Sqrt[-5 + 2*x]) - (40
92968*Sqrt[11/39]*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])/(2319687747*Sqrt[(2 - 3*x)/(5 -
2*x)]*Sqrt[7 + 5*x]) + (111628*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[
1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(74828637*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. \[ \int \frac{\sqrt{- 3 x + 2} \sqrt{4 x + 1}}{\sqrt{2 x - 5} \left (5 x + 7\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 2.00434, size = 251, normalized size = 0.76 \[ -\frac{2 \sqrt{2 x-5} \sqrt{4 x+1} \left (958111 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^3 F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-2046484 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^3 E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+31 \sqrt{\frac{5 x+7}{3 x-2}} \left (370051256 x^4+643813106 x^3-2953846743 x^2-2271416114 x-374624540\right )\right )}{90467822133 \sqrt{2-3 x} (5 x+7)^{5/2} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(Sqrt[-5 + 2*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)]*(-374624540 - 22
71416114*x - 2953846743*x^2 + 643813106*x^3 + 370051256*x^4) - 2046484*Sqrt[682]
*(-2 + 3*x)*(7 + 5*x)^3*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[ArcSin[S
qrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + 958111*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]*Sqr
t[(-5 + 2*x)/(-2 + 3*x)]], 39/62]))/(90467822133*Sqrt[2 - 3*x]*(7 + 5*x)^(5/2)*S
qrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2))

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Maple [B]  time = 0.039, 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)*(1+4*x)^(1/2)/(7+5*x)^(7/2)/(-5+2*x)^(1/2),x)

[Out]

-2/90467822133*(126500000*((-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^4-818593600*((-
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)*1
1^(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+417450000*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^3-2701358880*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+432946250*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))-2801636596*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(1/3
1*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))-945475608*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*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)*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))-100277716*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))-5843757936*x^4-10390893586*x^3+65568669813*x^
2+3127552098*x-26993559920)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(2-3*x)^(1/2)/(120*x^4-
182*x^3-385*x^2+197*x+70)/(7+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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