3.2944 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{3946 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{779 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{124 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{16732 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}+\frac{3946 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]

[Out]

(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (124*Sqrt[3 + 5*x])/(1
47*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (779*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 +
3*x)^(5/2)) - (2264*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2 + 3*x)^(3/2)) - (394
6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*Sqrt[2 + 3*x]) + (3946*Sqrt[11/3]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035 - (16732*Sqrt[3/11]*EllipticF[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035

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Rubi [A]  time = 0.518239, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{3946 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 \sqrt{3 x+2}}-\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{3/2}}-\frac{779 \sqrt{1-2 x} \sqrt{5 x+3}}{1715 (3 x+2)^{5/2}}+\frac{124 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{16732 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035}+\frac{3946 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{84035} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (124*Sqrt[3 + 5*x])/(1
47*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (779*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1715*(2 +
3*x)^(5/2)) - (2264*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2 + 3*x)^(3/2)) - (394
6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*Sqrt[2 + 3*x]) + (3946*Sqrt[11/3]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035 - (16732*Sqrt[3/11]*EllipticF[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/84035

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Rubi in Sympy [A]  time = 46.2316, size = 201, normalized size = 0.91 \[ - \frac{3946 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84035 \sqrt{3 x + 2}} - \frac{2264 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12005 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{3946 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{252105} - \frac{16732 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{924385} + \frac{1558 \sqrt{5 x + 3}}{5145 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{53 \sqrt{5 x + 3}}{245 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{11 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3946*sqrt(-2*x + 1)*sqrt(5*x + 3)/(84035*sqrt(3*x + 2)) - 2264*sqrt(-2*x + 1)*s
qrt(5*x + 3)/(12005*(3*x + 2)**(3/2)) + 3946*sqrt(33)*elliptic_e(asin(sqrt(21)*s
qrt(-2*x + 1)/7), 35/33)/252105 - 16732*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-
2*x + 1)/7), 35/33)/924385 + 1558*sqrt(5*x + 3)/(5145*sqrt(-2*x + 1)*(3*x + 2)**
(3/2)) - 53*sqrt(5*x + 3)/(245*sqrt(-2*x + 1)*(3*x + 2)**(5/2)) + 11*sqrt(5*x +
3)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2))

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Mathematica [A]  time = 0.345098, size = 108, normalized size = 0.49 \[ \frac{2 \left (\frac{\sqrt{5 x+3} \left (-213084 x^4-356292 x^3+2199 x^2+158902 x+43881\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}+\sqrt{2} \left (39620 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1973 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{252105} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[3 + 5*x]*(43881 + 158902*x + 2199*x^2 - 356292*x^3 - 213084*x^4))/((1
- 2*x)^(3/2)*(2 + 3*x)^(5/2)) + Sqrt[2]*(-1973*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2] + 39620*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/
252105

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Maple [C]  time = 0.036, size = 502, normalized size = 2.3 \[ -{\frac{2}{252105\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 713160\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-35514\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+594300\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-29595\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-158480\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+7892\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-158480\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +7892\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1065420\,{x}^{5}+2420712\,{x}^{4}+1057881\,{x}^{3}-801107\,{x}^{2}-696111\,x-131643 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/252105*(1-2*x)^(1/2)*(713160*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1
/2)-35514*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+594300*2^(1/2)*Ell
ipticF(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)-29595*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)-158480*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)+7892
*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)-158480*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))+7892*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))+1
065420*x^5+2420712*x^4+1057881*x^3-801107*x^2-696111*x-131643)/(2+3*x)^(5/2)/(-1
+2*x)^2/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((5*x + 3)^(3/2)/((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*sqrt(3
*x + 2)*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((3+5*x)**(3/2)/(1-2*x)**(5/2)/(2+3*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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