3.2919 \(\int \frac{(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{5/2}}{1815 (5 x+3)^{3/2}}-\frac{4421 \sqrt{1-2 x} (3 x+2)^{3/2}}{99825 \sqrt{5 x+3}}+\frac{83093 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{166375}+\frac{84134 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{75625 \sqrt{33}}+\frac{5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{151250 \sqrt{33}} \]

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(7/2)
)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4421*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(998
25*Sqrt[3 + 5*x]) + (83093*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/166375 + (
5684677*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(151250*Sqrt[33]) + (
84134*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(75625*Sqrt[33])

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Rubi [A]  time = 0.414265, 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 \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{5/2}}{1815 (5 x+3)^{3/2}}-\frac{4421 \sqrt{1-2 x} (3 x+2)^{3/2}}{99825 \sqrt{5 x+3}}+\frac{83093 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{166375}+\frac{84134 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{75625 \sqrt{33}}+\frac{5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{151250 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(7/2)
)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4421*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(998
25*Sqrt[3 + 5*x]) + (83093*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/166375 + (
5684677*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(151250*Sqrt[33]) + (
84134*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(75625*Sqrt[33])

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Rubi in Sympy [A]  time = 39.6059, size = 172, normalized size = 0.92 \[ - \frac{107 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{1815 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{4421 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{99825 \sqrt{5 x + 3}} + \frac{83093 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{166375} + \frac{5684677 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{4991250} + \frac{84134 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2646875} + \frac{7 \left (3 x + 2\right )^{\frac{7}{2}}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

-107*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(1815*(5*x + 3)**(3/2)) - 4421*sqrt(-2*x +
1)*(3*x + 2)**(3/2)/(99825*sqrt(5*x + 3)) + 83093*sqrt(-2*x + 1)*sqrt(3*x + 2)*s
qrt(5*x + 3)/166375 + 5684677*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7
), 35/33)/4991250 + 84134*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11),
33/35)/2646875 + 7*(3*x + 2)**(7/2)/(11*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.263221, size = 136, normalized size = 0.73 \[ \frac{10 \sqrt{3 x+2} \left (-2695275 x^3+9376775 x^2+14153413 x+4534181\right ) \sqrt{5 x+3}+2908255 \sqrt{2-4 x} (5 x+3)^2 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5684677 \sqrt{2-4 x} (5 x+3)^2 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{4991250 \sqrt{1-2 x} (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(4534181 + 14153413*x + 9376775*x^2 - 2695275*x^
3) - 5684677*Sqrt[2 - 4*x]*(3 + 5*x)^2*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2] + 2908255*Sqrt[2 - 4*x]*(3 + 5*x)^2*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])/(4991250*Sqrt[1 - 2*x]*(3 + 5*x)^2)

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Maple [C]  time = 0.033, size = 272, normalized size = 1.5 \[ -{\frac{1}{29947500\,{x}^{2}+4991250\,x-9982500}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 14541275\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-28423385\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8724765\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -17054031\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -80858250\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+90683620 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(5/2),x)

[Out]

-1/4991250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(14541275*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*(3+5*x)^(1/2)*(2+3*x)^(
1/2)*(1-2*x)^(1/2)-28423385*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)+87
24765*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))-17054031*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))-80858250*x^4+227397750*x^3+612137890*x^2+41909369
0*x+90683620)/(3+5*x)^(3/2)/(6*x^2+x-2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

integral(-(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(3*x + 2)/((50*x^3 + 35*x
^2 - 12*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)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)), x)