3.2707 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{135 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{9808 \sqrt{1-2 x} (5 x+3)^{3/2}}{945 \sqrt{3 x+2}}-\frac{43214 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1701}-\frac{43214 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505} \]

[Out]

(-43214*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1701 + (9808*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(945*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(15*(2
+ 3*x)^(5/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(135*(2 + 3*x)^(3/2)) + (116
854*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8505 - (43214*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8505

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Rubi [A]  time = 0.411239, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{135 (3 x+2)^{3/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac{9808 \sqrt{1-2 x} (5 x+3)^{3/2}}{945 \sqrt{3 x+2}}-\frac{43214 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{1701}-\frac{43214 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8505} \]

Antiderivative was successfully verified.

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

[Out]

(-43214*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1701 + (9808*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(945*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(15*(2
+ 3*x)^(5/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(135*(2 + 3*x)^(3/2)) + (116
854*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8505 - (43214*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/8505

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Rubi in Sympy [A]  time = 40.4672, size = 172, normalized size = 0.9 \[ - \frac{394 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{189 \sqrt{3 x + 2}} - \frac{362 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{945 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{4208 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1701} + \frac{116854 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{25515} - \frac{43214 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{25515} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-394*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(189*sqrt(3*x + 2)) - 362*(-2*x + 1)**(3/2)
*(5*x + 3)**(3/2)/(945*(3*x + 2)**(3/2)) - 2*(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/
(15*(3*x + 2)**(5/2)) - 4208*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1701 + 1
16854*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/25515 - 43214*
sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/25515

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Mathematica [A]  time = 0.34056, size = 104, normalized size = 0.54 \[ \frac{\sqrt{2} \left (829885 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-116854 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (47250 x^3+377793 x^2+432387 x+134497\right )}{(3 x+2)^{5/2}}}{25515} \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(134497 + 432387*x + 377793*x^2 + 47250*x^3))/(
2 + 3*x)^(5/2) + Sqrt[2]*(-116854*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2] + 829885*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/25515

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Maple [C]  time = 0.029, size = 391, normalized size = 2.1 \[ -{\frac{1}{255150\,{x}^{2}+25515\,x-76545} \left ( 7468965\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1051686\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+9958620\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1402248\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3319540\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -467416\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2835000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}+3863868\,{x}^{2}-6975984\,x-2420946 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25515*(7468965*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)-1051686*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)+9958620*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)-1402248*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)+3319540*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-467416*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))+2835000*x^5+22951080*x^4+27359478*x^3+38638
68*x^2-6975984*x-2420946)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((27*x^3 + 5
4*x^2 + 36*x + 8)*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)**(3/2)*(3+5*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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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