3.2789 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{524}{225} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{442 (1-2 x)^{3/2} (3 x+2)^{7/2}}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac{59662 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{7875}+\frac{373022 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{196875}+\frac{500501 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{984375}-\frac{595387 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875}-\frac{1065118 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (442*(1 - 2*x)^(3/2)
*(2 + 3*x)^(7/2))/(75*Sqrt[3 + 5*x]) + (500501*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x])/984375 + (373022*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/196875 +
(59662*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/7875 - (524*Sqrt[1 - 2*x]*(
2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/225 - (1065118*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/4921875 - (595387*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/4921875

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Rubi [A]  time = 0.581851, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{524}{225} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{442 (1-2 x)^{3/2} (3 x+2)^{7/2}}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac{59662 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{7875}+\frac{373022 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{196875}+\frac{500501 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{984375}-\frac{595387 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875}-\frac{1065118 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (442*(1 - 2*x)^(3/2)
*(2 + 3*x)^(7/2))/(75*Sqrt[3 + 5*x]) + (500501*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[
3 + 5*x])/984375 + (373022*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/196875 +
(59662*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/7875 - (524*Sqrt[1 - 2*x]*(
2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/225 - (1065118*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/4921875 - (595387*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/4921875

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Rubi in Sympy [A]  time = 58.2501, size = 230, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{7}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{442 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}}}{825 \sqrt{5 x + 3}} - \frac{212 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{825} + \frac{2264 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2625} + \frac{3863 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{21875} + \frac{94886 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{984375} - \frac{1065118 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{14765625} - \frac{6549257 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{172265625} \]

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)**(5/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**(7/2)/(15*(5*x + 3)**(3/2)) - 442*(-2*x + 1)**(5
/2)*(3*x + 2)**(5/2)/(825*sqrt(5*x + 3)) - 212*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2
)*sqrt(5*x + 3)/825 + 2264*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/2625
+ 3863*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/21875 + 94886*sqrt(-2*x +
1)*sqrt(3*x + 2)*sqrt(5*x + 3)/984375 - 1065118*sqrt(33)*elliptic_e(asin(sqrt(21
)*sqrt(-2*x + 1)/7), 35/33)/14765625 - 6549257*sqrt(35)*elliptic_f(asin(sqrt(55)
*sqrt(-2*x + 1)/11), 33/35)/172265625

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Mathematica [A]  time = 0.487237, size = 117, normalized size = 0.46 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (4725000 x^5+1327500 x^4-5654250 x^3+470675 x^2+4026600 x+1215489\right )}{(5 x+3)^{3/2}}[phone] \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2130236 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{29531250} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(1215489 + 4026600*x + 470675*x^2 - 5654250*x^3
+ 1327500*x^4 + 4725000*x^5))/(3 + 5*x)^(3/2) + 2130236*Sqrt[2]*EllipticE[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 17517535*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]], -33/2])/29531250

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Maple [C]  time = 0.029, size = 287, normalized size = 1.1 \[ -{\frac{1}{177187500\,{x}^{2}[phone]\,x-59062500} \left ( -850500000\,{x}^{7}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-380700000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +6390708\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-1078163250\,{x}^{3}-311345520\,{x}^{2}[phone]\,x+72929340 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

-1/29531250*(-850500000*x^7+87587675*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)[phone]*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)-380700000*x^6+
52552605*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))+6390708*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^5+164556000*x^4-1078163250*x^3-311
345520*x^2+205131330*x+72929340)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)
^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*sqrt(3*x + 2)*sqrt(-2*x
+ 1)/((25*x^2 + 30*x + 9)*sqrt(5*x + 3)), 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)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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