3.2976 \(\int \frac{(2+3 x)^{7/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{63 (3 x+2)^{3/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{29933 \sqrt{1-2 x} \sqrt{3 x+2}}{219615 \sqrt{5 x+3}}+\frac{908 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 (5 x+3)^{3/2}}-\frac{1847 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}}-\frac{29933 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}} \]

[Out]

(908*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(19965*(3 + 5*x)^(3/2)) - (63*(2 + 3*x)^(3/2))
/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(5/2))/(33*(1 - 2*x)^(3/2)*(
3 + 5*x)^(3/2)) + (29933*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(219615*Sqrt[3 + 5*x]) - (
29933*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(33275*Sqrt[33]) - (184
7*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(33275*Sqrt[33])

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Rubi [A]  time = 0.426841, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 (3 x+2)^{5/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{63 (3 x+2)^{3/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{29933 \sqrt{1-2 x} \sqrt{3 x+2}}{219615 \sqrt{5 x+3}}+\frac{908 \sqrt{1-2 x} \sqrt{3 x+2}}{19965 (5 x+3)^{3/2}}-\frac{1847 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}}-\frac{29933 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{33275 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(908*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(19965*(3 + 5*x)^(3/2)) - (63*(2 + 3*x)^(3/2))
/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(5/2))/(33*(1 - 2*x)^(3/2)*(
3 + 5*x)^(3/2)) + (29933*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(219615*Sqrt[3 + 5*x]) - (
29933*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(33275*Sqrt[33]) - (184
7*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(33275*Sqrt[33])

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Rubi in Sympy [A]  time = 38.9856, size = 172, normalized size = 0.92 \[ \frac{29933 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{219615 \sqrt{5 x + 3}} + \frac{908 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{19965 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{29933 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1098075} - \frac{1847 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1098075} - \frac{63 \left (3 x + 2\right )^{\frac{3}{2}}}{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \left (3 x + 2\right )^{\frac{5}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

29933*sqrt(-2*x + 1)*sqrt(3*x + 2)/(219615*sqrt(5*x + 3)) + 908*sqrt(-2*x + 1)*s
qrt(3*x + 2)/(19965*(5*x + 3)**(3/2)) - 29933*sqrt(33)*elliptic_e(asin(sqrt(21)*
sqrt(-2*x + 1)/7), 35/33)/1098075 - 1847*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(
-2*x + 1)/7), 35/33)/1098075 - 63*(3*x + 2)**(3/2)/(121*sqrt(-2*x + 1)*(5*x + 3)
**(3/2)) + 7*(3*x + 2)**(5/2)/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.312553, size = 107, normalized size = 0.57 \[ \frac{\frac{10 \sqrt{3 x+2} \left (598660 x^3+905823 x^2+423882 x+57437\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}+1085 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+59866 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2196150} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[2 + 3*x]*(57437 + 423882*x + 905823*x^2 + 598660*x^3))/((1 - 2*x)^(3/2
)*(3 + 5*x)^(3/2)) + 59866*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2] + 1085*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/2196150

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Maple [C]  time = 0.036, size = 383, normalized size = 2.1 \[ -{\frac{1}{2196150\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 10850\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+598660\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1085\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+59866\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3255\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -179598\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -17959800\,{x}^{4}-39147890\,{x}^{3}-30832920\,{x}^{2}-10200750\,x-1148740 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2196150*(1-2*x)^(1/2)*(10850*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^
(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1
/2)+598660*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*
3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+1085*2^(1/2)*Elli
pticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5
*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+59866*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1
/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*
(1-2*x)^(1/2)-3255*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1
/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-179598*2^(1/2
)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5
*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-17959800*x^4-39147890*x^3-30832920*x^2
-10200750*x-1148740)/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x + 2)/((100*x^4 + 20*x^3 - 59*x^2
- 6*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)), 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)**(7/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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