3.2973 \(\int \frac{(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{294 (3 x+2)^{9/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4373 \sqrt{1-2 x} (3 x+2)^{7/2}}{19965 (5 x+3)^{3/2}}+\frac{150812 \sqrt{1-2 x} (3 x+2)^{5/2}}{1098075 \sqrt{5 x+3}}-\frac{31887029 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{18301250}-\frac{371279941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{45753125}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41593750 \sqrt{33}}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83187500 \sqrt{33}} \]

[Out]

(4373*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(19965*(3 + 5*x)^(3/2)) - (294*(2 + 3*x)^(9
/2))/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(11/2))/(33*(1 - 2*x)^(3
/2)*(3 + 5*x)^(3/2)) + (150812*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(1098075*Sqrt[3 +
5*x]) - (371279941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45753125 - (318870
29*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/18301250 - (51601293223*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(83187500*Sqrt[33]) - (776112041*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41593750*Sqrt[33])

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Rubi [A]  time = 0.584363, antiderivative size = 249, 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 \[ \frac{7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{294 (3 x+2)^{9/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4373 \sqrt{1-2 x} (3 x+2)^{7/2}}{19965 (5 x+3)^{3/2}}+\frac{150812 \sqrt{1-2 x} (3 x+2)^{5/2}}{1098075 \sqrt{5 x+3}}-\frac{31887029 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{18301250}-\frac{371279941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{45753125}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41593750 \sqrt{33}}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83187500 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^(13/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(4373*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(19965*(3 + 5*x)^(3/2)) - (294*(2 + 3*x)^(9
/2))/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(11/2))/(33*(1 - 2*x)^(3
/2)*(3 + 5*x)^(3/2)) + (150812*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(1098075*Sqrt[3 +
5*x]) - (371279941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45753125 - (318870
29*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/18301250 - (51601293223*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(83187500*Sqrt[33]) - (776112041*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41593750*Sqrt[33])

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Rubi in Sympy [A]  time = 56.7269, size = 230, normalized size = 0.92 \[ \frac{4373 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{19965 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{150812 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{1098075 \sqrt{5 x + 3}} - \frac{31887029 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{18301250} - \frac{371279941 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{45753125} - \frac{51601293223 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2745187500} - \frac{776112041 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1455781250} - \frac{294 \left (3 x + 2\right )^{\frac{9}{2}}}{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \left (3 x + 2\right )^{\frac{11}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(13/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

4373*sqrt(-2*x + 1)*(3*x + 2)**(7/2)/(19965*(5*x + 3)**(3/2)) + 150812*sqrt(-2*x
+ 1)*(3*x + 2)**(5/2)/(1098075*sqrt(5*x + 3)) - 31887029*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*sqrt(5*x + 3)/18301250 - 371279941*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5
*x + 3)/45753125 - 51601293223*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/
7), 35/33)/2745187500 - 776112041*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x +
1)/11), 33/35)/1455781250 - 294*(3*x + 2)**(9/2)/(121*sqrt(-2*x + 1)*(5*x + 3)**
(3/2)) + 7*(3*x + 2)**(11/2)/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.420166, size = 117, normalized size = 0.47 \[ \frac{-\frac{10 \sqrt{3 x+2} \left (8004966750 x^5+53010668700 x^4-222254370925 x^3-215557803774 x^2+21979664649 x+36533948644\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}-25989595870 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2745187500} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^(13/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

((-10*Sqrt[2 + 3*x]*(36533948644 + 21979664649*x - 215557803774*x^2 - 2222543709
25*x^3 + 53010668700*x^4 + 8004966750*x^5))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) +
51601293223*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2598959
5870*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/2745187500

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Maple [C]  time = 0.038, size = 393, normalized size = 1.6 \[{\frac{1}{2745187500\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 259895958700\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-516012932230\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-51601293223\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-240149002500\,{x}^{6}-77968787610\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1750419396000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-1535611752300\,x-730678972880 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(13/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/2745187500*(1-2*x)^(1/2)*(259895958700*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)-516012932230*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)+25
989595870*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)-51601293223*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)-240149002500*x^6-77968787610*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))[phone]*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))-1750419396000*x^5+5607417753750*x^4+10911821531720*x^3+36517
66136010*x^2-1535611752300*x-730678972880)/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*x)^(1/2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(13/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(13/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{3 \, x + 2}}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(13/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="fricas")

[Out]

integral((729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*sqrt
(3*x + 2)/((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*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((2+3*x)**(13/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(13/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(13/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)