3.2955 \(\int \frac{(2+3 x)^{7/2}}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=156 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{33 (1-2 x)^{3/2}}-\frac{679 \sqrt{5 x+3} (3 x+2)^{3/2}}{363 \sqrt{1-2 x}}-\frac{4517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1210}-\frac{4721 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{550 \sqrt{33}}-\frac{78472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}} \]

[Out]

(-4517*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1210 - (679*(2 + 3*x)^(3/2)*Sq
rt[3 + 5*x])/(363*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(33*(1 - 2*
x)^(3/2)) - (78472*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(275*Sqrt[
33]) - (4721*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(550*Sqrt[33])

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Rubi [A]  time = 0.332968, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{5/2}}{33 (1-2 x)^{3/2}}-\frac{679 \sqrt{5 x+3} (3 x+2)^{3/2}}{363 \sqrt{1-2 x}}-\frac{4517 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1210}-\frac{4721 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{550 \sqrt{33}}-\frac{78472 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-4517*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1210 - (679*(2 + 3*x)^(3/2)*Sq
rt[3 + 5*x])/(363*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(33*(1 - 2*
x)^(3/2)) - (78472*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(275*Sqrt[
33]) - (4721*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(550*Sqrt[33])

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Rubi in Sympy [A]  time = 31.0858, size = 143, normalized size = 0.92 \[ - \frac{4517 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1210} - \frac{78472 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9075} - \frac{4721 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{19250} - \frac{679 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

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)**(1/2),x)

[Out]

-4517*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1210 - 78472*sqrt(33)*elliptic_
e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9075 - 4721*sqrt(35)*elliptic_f(asin(s
qrt(55)*sqrt(-2*x + 1)/11), 33/35)/19250 - 679*(3*x + 2)**(3/2)*sqrt(5*x + 3)/(3
63*sqrt(-2*x + 1)) + 7*(3*x + 2)**(5/2)*sqrt(5*x + 3)/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.320229, size = 120, normalized size = 0.77 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (6534 x^2-70234 x+24051\right )-158095 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+313888 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{36300 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(24051 - 70234*x + 6534*x^2) + 313888*Sqrt[2 -
4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 158095*Sqrt
[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(36300*
(1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 281, normalized size = 1.8 \[{\frac{1}{36300\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 316190\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-627776\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-158095\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +313888\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -980100\,{x}^{4}+9293640\,{x}^{3}+9344770\,{x}^{2}-355650\,x-1443060 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \]

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)^(1/2),x)

[Out]

1/36300*(316190*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)-627776*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)-158095*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))+313888*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticE(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))-980100*
x^4+9293640*x^3+9344770*x^2-355650*x-1443060)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x
)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x + 2)/((4*x^2 - 4*x + 1)*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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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