3.2775 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{16564 \sqrt{5 x+3} \sqrt{1-2 x}}{135 \sqrt{3 x+2}}+\frac{1736 \sqrt{5 x+3} \sqrt{1-2 x}}{135 (3 x+2)^{3/2}}-\frac{496}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{16564}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (1736*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/(135*(2 + 3*x)^(3/2)) + (16564*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(135*Sq
rt[2 + 3*x]) - (16564*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3
3])/135 - (496*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/135

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Rubi [A]  time = 0.338556, antiderivative size = 160, 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 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}+\frac{16564 \sqrt{5 x+3} \sqrt{1-2 x}}{135 \sqrt{3 x+2}}+\frac{1736 \sqrt{5 x+3} \sqrt{1-2 x}}{135 (3 x+2)^{3/2}}-\frac{496}{135} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{16564}{135} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (1736*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/(135*(2 + 3*x)^(3/2)) + (16564*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(135*Sq
rt[2 + 3*x]) - (16564*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3
3])/135 - (496*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/135

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Rubi in Sympy [A]  time = 33.1561, size = 143, normalized size = 0.89 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{16564 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{135 \sqrt{3 x + 2}} + \frac{1736 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{135 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{16564 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{405} - \frac{5456 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{4725} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(15*(3*x + 2)**(5/2)) + 16564*sqrt(-2*x + 1)*
sqrt(5*x + 3)/(135*sqrt(3*x + 2)) + 1736*sqrt(-2*x + 1)*sqrt(5*x + 3)/(135*(3*x
+ 2)**(3/2)) - 16564*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)
/405 - 5456*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/4725

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Mathematica [A]  time = 0.313951, size = 101, normalized size = 0.63 \[ \frac{4}{405} \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (74538 x^2+101862 x+34927\right )}{2 (3 x+2)^{5/2}}+\sqrt{2} \left (4141 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2095 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(34927 + 101862*x + 74538*x^2))/(2*(2 + 3*x)^
(5/2)) + Sqrt[2]*(4141*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2095
*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/405

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Maple [C]  time = 0.03, size = 386, normalized size = 2.4 \[{\frac{2}{4050\,{x}^{2}+405\,x-1215} \left ( 37710\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-74538\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+50280\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-99384\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+16760\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -33128\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2236140\,{x}^{4}+3279474\,{x}^{3}+682554\,{x}^{2}-811977\,x-314343 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/405*(37710*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)-74538*2^(1/2)*E
llipticE(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)+50280*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)-99384*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)+16760
*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))-33128*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))+2236140*x^4+3279474*x^3+682554*x^2-811977*x-314343)*(3+5*
x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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