3.2706 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{614}{27} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{2632}{243} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{2632 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215}-\frac{9587 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215} \]

[Out]

(2632*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/243 - (614*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/27 - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(
3/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(27*Sqrt[2 + 3*x]) - (9587*Sqrt[11/3
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215 + (2632*Sqrt[11/3]*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215

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Rubi [A]  time = 0.413511, 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 \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{614}{27} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{2632}{243} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{2632 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215}-\frac{9587 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1215} \]

Antiderivative was successfully verified.

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

[Out]

(2632*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/243 - (614*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/27 - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(
3/2)) + (362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(27*Sqrt[2 + 3*x]) - (9587*Sqrt[11/3
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215 + (2632*Sqrt[11/3]*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1215

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Rubi in Sympy [A]  time = 40.841, size = 172, normalized size = 0.9 \[ - \frac{362 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{189 \sqrt{3 x + 2}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{9 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{316 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{189} + \frac{2632 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{243} - \frac{9587 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3645} + \frac{2632 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3645} \]

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)**(5/2),x)

[Out]

-362*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/(189*sqrt(3*x + 2)) - 2*(-2*x + 1)**(3/2
)*(5*x + 3)**(5/2)/(9*(3*x + 2)**(3/2)) - 316*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x
+ 3)**(3/2)/189 + 2632*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/243 - 9587*sqr
t(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3645 + 2632*sqrt(33)*el
liptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3645

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Mathematica [A]  time = 0.417023, size = 107, normalized size = 0.56 \[ \frac{-\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (810 x^3-468 x^2-2463 x-1187\right )}{(3 x+2)^{3/2}}-53015 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+9587 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{3645} \]

Antiderivative was successfully verified.

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

[Out]

((-30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-1187 - 2463*x - 468*x^2 + 810*x^3))/(2 + 3*x
)^(3/2) + 9587*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5301
5*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/3645

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Maple [C]  time = 0.03, size = 277, normalized size = 1.5 \[{\frac{1}{36450\,{x}^{2}+3645\,x-10935} \left ( 159045\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-28761\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+106030\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -19174\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -243000\,{x}^{5}+116100\,{x}^{4}+825840\,{x}^{3}+387870\,{x}^{2}-186060\,x-106830 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

1/3645*(159045*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)-28761*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*(
3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+106030*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))-19174*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elliptic
E(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))-243000*x^5
+116100*x^4+825840*x^3+387870*x^2-186060*x-106830)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(
10*x^2+x-3)/(2+3*x)^(3/2)

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

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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((9*x^2 + 12
*x + 4)*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)**(5/2),x)

[Out]

Timed out

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

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)^(5/2),x, algorithm="giac")

[Out]

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