3.1045 \(\int (2-5 x) x^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=233 \[ \frac{556 \sqrt{x} \left (3 x^2+5 x+2\right )^{5/2}}{1287}-\frac{4 \sqrt{x} (8575 x+6959) \left (3 x^2+5 x+2\right )^{3/2}}{27027}+\frac{8 \sqrt{x} (6381 x+6908) \sqrt{3 x^2+5 x+2}}{243243}+\frac{55112 \sqrt{x} (3 x+2)}{729729 \sqrt{3 x^2+5 x+2}}+\frac{25448 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243243 \sqrt{3 x^2+5 x+2}}-\frac{55112 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{729729 \sqrt{3 x^2+5 x+2}}-\frac{10}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2} \]

[Out]

(55112*Sqrt[x]*(2 + 3*x))/(729729*Sqrt[2 + 5*x + 3*x^2]) + (8*Sqrt[x]*(6908 + 63
81*x)*Sqrt[2 + 5*x + 3*x^2])/243243 - (4*Sqrt[x]*(6959 + 8575*x)*(2 + 5*x + 3*x^
2)^(3/2))/27027 + (556*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/1287 - (10*x^(3/2)*(2 +
5*x + 3*x^2)^(5/2))/39 - (55112*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elliptic
E[ArcTan[Sqrt[x]], -1/2])/(729729*Sqrt[2 + 5*x + 3*x^2]) + (25448*Sqrt[2]*(1 + x
)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(243243*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi [A]  time = 0.404525, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{556 \sqrt{x} \left (3 x^2+5 x+2\right )^{5/2}}{1287}-\frac{4 \sqrt{x} (8575 x+6959) \left (3 x^2+5 x+2\right )^{3/2}}{27027}+\frac{8 \sqrt{x} (6381 x+6908) \sqrt{3 x^2+5 x+2}}{243243}+\frac{55112 \sqrt{x} (3 x+2)}{729729 \sqrt{3 x^2+5 x+2}}+\frac{25448 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243243 \sqrt{3 x^2+5 x+2}}-\frac{55112 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{729729 \sqrt{3 x^2+5 x+2}}-\frac{10}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(55112*Sqrt[x]*(2 + 3*x))/(729729*Sqrt[2 + 5*x + 3*x^2]) + (8*Sqrt[x]*(6908 + 63
81*x)*Sqrt[2 + 5*x + 3*x^2])/243243 - (4*Sqrt[x]*(6959 + 8575*x)*(2 + 5*x + 3*x^
2)^(3/2))/27027 + (556*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/1287 - (10*x^(3/2)*(2 +
5*x + 3*x^2)^(5/2))/39 - (55112*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elliptic
E[ArcTan[Sqrt[x]], -1/2])/(729729*Sqrt[2 + 5*x + 3*x^2]) + (25448*Sqrt[2]*(1 + x
)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(243243*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi in Sympy [A]  time = 43.4578, size = 221, normalized size = 0.95 \[ - \frac{10 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{39} + \frac{27556 \sqrt{x} \left (6 x + 4\right )}{729729 \sqrt{3 x^{2} + 5 x + 2}} - \frac{16 \sqrt{x} \left (\frac{77175 x}{4} + \frac{62631}{4}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{243243} + \frac{32 \sqrt{x} \left (\frac{95715 x}{4} + 25905\right ) \sqrt{3 x^{2} + 5 x + 2}}{3648645} + \frac{556 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{1287} - \frac{13778 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{729729 \sqrt{3 x^{2} + 5 x + 2}} + \frac{6362 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{243243 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*x**(3/2)*(3*x**2 + 5*x + 2)**(5/2)/39 + 27556*sqrt(x)*(6*x + 4)/(729729*sqrt
(3*x**2 + 5*x + 2)) - 16*sqrt(x)*(77175*x/4 + 62631/4)*(3*x**2 + 5*x + 2)**(3/2)
/243243 + 32*sqrt(x)*(95715*x/4 + 25905)*sqrt(3*x**2 + 5*x + 2)/3648645 + 556*sq
rt(x)*(3*x**2 + 5*x + 2)**(5/2)/1287 - 13778*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*e
lliptic_e(atan(sqrt(x)), -1/2)/(729729*sqrt(3*x**2 + 5*x + 2)) + 6362*sqrt((6*x
+ 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(243243*sqrt(3*x**2 + 5*
x + 2))

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Mathematica [C]  time = 0.268264, size = 178, normalized size = 0.76 \[ \frac{21232 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+55112 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (2525985 x^8+8374023 x^7+8989785 x^6+1830195 x^5-2497986 x^4-1171602 x^3+8508 x^2-61436 x-55112\right )}{729729 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-55112 - 61436*x + 8508*x^2 - 1171602*x^3 - 2497986*x^4 + 1830195*x^5 + 898
9785*x^6 + 8374023*x^7 + 2525985*x^8) + (55112*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[
3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (21232*I)*Sqrt[2
]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]],
3/2])/(729729*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.015, size = 143, normalized size = 0.6 \[ -{\frac{2}{2189187} \left ( 7577955\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}+5490585\,{x}^{5}+3162\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -13778\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -7493958\,{x}^{4}-3514806\,{x}^{3}+273528\,{x}^{2}+229032\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/2189187/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(7577955*x^8+25122069*x^7+26969355*x^6+54
90585*x^5+3162*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*EllipticF(
1/2*(6*x+4)^(1/2),I*2^(1/2))-13778*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(
-x)^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))-7493958*x^4-3514806*x^3+273528*
x^2+229032*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (15 \, x^{4} + 19 \, x^{3} - 4 \, x\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(15*x^4 + 19*x^3 - 4*x)*sqrt(3*x^2 + 5*x + 2)*sqrt(x), 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((2-5*x)*x**(3/2)*(3*x**2+5*x+2)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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