3.2920 \(\int \frac{(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ \frac{7 (3 x+2)^{5/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{3/2}}{1815 (5 x+3)^{3/2}}-\frac{4289 \sqrt{1-2 x} \sqrt{3 x+2}}{99825 \sqrt{5 x+3}}+\frac{2657 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}}+\frac{118898 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}} \]

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(5/2)
)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4289*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(99825
*Sqrt[3 + 5*x]) + (118898*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15
125*Sqrt[33]) + (2657*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15125*
Sqrt[33])

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Rubi [A]  time = 0.340687, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{7 (3 x+2)^{5/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{3/2}}{1815 (5 x+3)^{3/2}}-\frac{4289 \sqrt{1-2 x} \sqrt{3 x+2}}{99825 \sqrt{5 x+3}}+\frac{2657 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}}+\frac{118898 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15125 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(5/2)
)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4289*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(99825
*Sqrt[3 + 5*x]) + (118898*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15
125*Sqrt[33]) + (2657*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15125*
Sqrt[33])

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Rubi in Sympy [A]  time = 32.058, size = 143, normalized size = 0.92 \[ - \frac{107 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{1815 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{4289 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{99825 \sqrt{5 x + 3}} + \frac{118898 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{499125} + \frac{2657 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{529375} + \frac{7 \left (3 x + 2\right )^{\frac{5}{2}}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107*sqrt(-2*x + 1)*(3*x + 2)**(3/2)/(1815*(5*x + 3)**(3/2)) - 4289*sqrt(-2*x +
1)*sqrt(3*x + 2)/(99825*sqrt(5*x + 3)) + 118898*sqrt(33)*elliptic_e(asin(sqrt(21
)*sqrt(-2*x + 1)/7), 35/33)/499125 + 2657*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt
(-2*x + 1)/11), 33/35)/529375 + 7*(3*x + 2)**(5/2)/(11*sqrt(-2*x + 1)*(5*x + 3)*
*(3/2))

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Mathematica [A]  time = 0.212499, size = 102, normalized size = 0.65 \[ \frac{\frac{10 \sqrt{3 x+2} \left (649925 x^2+772474 x+229463\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}+150115 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-237796 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{998250} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[2 + 3*x]*(229463 + 772474*x + 649925*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^(3
/2)) - 237796*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 15011
5*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/998250

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Maple [C]  time = 0.035, size = 267, normalized size = 1.7 \[ -{\frac{1}{5989500\,{x}^{2}+998250\,x-1996500}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 750575\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1188980\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+450345\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -713388\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+4589260 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/998250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(750575*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)-1188980*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)+450345
*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))-713388*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))[phone]*x^3+36172720*x^2+22333370*x+4589260)/(3+5*x)^(3
/2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2}}{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(27*x^3 + 54*x^2 + 36*x + 8)*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)**(7/2)/(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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