3.359 \(\int \frac{x^3 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=460 \[ \frac{\left (b c \left (e \left (2 d \sqrt{b^2-4 a c}-3 a e\right )+c d^2\right )+c \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )\right )-b^2 e \left (e \sqrt{b^2-4 a c}+2 c d\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (b c \left (c d^2-e \left (2 d \sqrt{b^2-4 a c}+3 a e\right )\right )-c \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )\right )-b^2 e \left (2 c d-e \sqrt{b^2-4 a c}\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x^2} (c d-b e)}{c^2}+\frac{\left (d+e x^2\right )^{3/2}}{3 c} \]

[Out]

((c*d - b*e)*Sqrt[d + e*x^2])/c^2 + (d + e*x^2)^(3/2)/(3*c) + ((b^3*e^2 - b^2*e*
(2*c*d + Sqrt[b^2 - 4*a*c]*e) + c*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a
*c]*d - 4*a*e)) + b*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 3*a*e)))*ArcTanh[(Sqrt
[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*
c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b^3*e^2 -
b^2*e*(2*c*d - Sqrt[b^2 - 4*a*c]*e) + b*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + 3
*a*e)) - c*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(S
qrt[2]*c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi [A]  time = 12.0647, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\left (b c \left (e \left (2 d \sqrt{b^2-4 a c}-3 a e\right )+c d^2\right )+c \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )\right )-b^2 e \left (e \sqrt{b^2-4 a c}+2 c d\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (b c \left (c d^2-e \left (2 d \sqrt{b^2-4 a c}+3 a e\right )\right )-c \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )\right )-b^2 e \left (2 c d-e \sqrt{b^2-4 a c}\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x^2} (c d-b e)}{c^2}+\frac{\left (d+e x^2\right )^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]

[Out]

((c*d - b*e)*Sqrt[d + e*x^2])/c^2 + (d + e*x^2)^(3/2)/(3*c) + ((b^3*e^2 - b^2*e*
(2*c*d + Sqrt[b^2 - 4*a*c]*e) + c*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a
*c]*d - 4*a*e)) + b*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 3*a*e)))*ArcTanh[(Sqrt
[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*
c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b^3*e^2 -
b^2*e*(2*c*d - Sqrt[b^2 - 4*a*c]*e) + b*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + 3
*a*e)) - c*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(S
qrt[2]*c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 1.70559, size = 448, normalized size = 0.97 \[ \frac{-\frac{3 \sqrt{2} \left (-b c \left (e \left (2 d \sqrt{b^2-4 a c}-3 a e\right )+c d^2\right )+c \left (c d \left (d \sqrt{b^2-4 a c}-4 a e\right )-a e^2 \sqrt{b^2-4 a c}\right )+b^2 e \left (e \sqrt{b^2-4 a c}+2 c d\right )+b^3 \left (-e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{3 \sqrt{2} \left (b c \left (c d^2-e \left (2 d \sqrt{b^2-4 a c}+3 a e\right )\right )+c \left (c d \left (d \sqrt{b^2-4 a c}+4 a e\right )-a e^2 \sqrt{b^2-4 a c}\right )+b^2 e \left (e \sqrt{b^2-4 a c}-2 c d\right )+b^3 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+2 \sqrt{c} \sqrt{d+e x^2} \left (-3 b e+4 c d+c e x^2\right )}{6 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(d + e*x^2)^(3/2))/(a + b*x^2 + c*x^4),x]

[Out]

(2*Sqrt[c]*Sqrt[d + e*x^2]*(4*c*d - 3*b*e + c*e*x^2) - (3*Sqrt[2]*(-(b^3*e^2) +
b^2*e*(2*c*d + Sqrt[b^2 - 4*a*c]*e) + c*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[
b^2 - 4*a*c]*d - 4*a*e)) - b*c*(c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - 3*a*e)))*ArcT
anh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/
(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) - (3*Sqrt[2]*(b^3*e
^2 + b^2*e*(-2*c*d + Sqrt[b^2 - 4*a*c]*e) + b*c*(c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*
d + 3*a*e)) + c*(-(a*Sqrt[b^2 - 4*a*c]*e^2) + c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e))
)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])
*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/(6*c^(5/2))

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Maple [C]  time = 0.041, size = 490, normalized size = 1.1 \[ -{\frac{{x}^{3}}{6\,c}{e}^{{\frac{3}{2}}}}+{\frac{e{x}^{2}}{8\,c}\sqrt{e{x}^{2}+d}}-{\frac{3\,dx}{4\,c}\sqrt{e}}+{\frac{1}{24\,c} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{bx}{2\,{c}^{2}}{e}^{{\frac{3}{2}}}}-{\frac{be}{2\,{c}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{5\,d}{8\,c}\sqrt{e{x}^{2}+d}}+{\frac{1}{4\,{c}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{6}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{4}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){{\it \_Z}}^{2}+c{d}^{4} \right ) }{\frac{ \left ( -ac{e}^{2}+{b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ){{\it \_R}}^{6}+ \left ( 4\,ab{e}^{3}-5\,acd{e}^{2}-3\,{b}^{2}d{e}^{2}+6\,bc{d}^{2}e-3\,{c}^{2}{d}^{3} \right ){{\it \_R}}^{4}+d \left ( -4\,ab{e}^{3}+5\,acd{e}^{2}+3\,{b}^{2}d{e}^{2}-6\,bc{d}^{2}e+3\,{c}^{2}{d}^{3} \right ){{\it \_R}}^{2}+ac{d}^{3}{e}^{2}-{b}^{2}{d}^{3}{e}^{2}+2\,bc{d}^{4}e-{c}^{2}{d}^{5}}{{{\it \_R}}^{7}c+3\,{{\it \_R}}^{5}be-3\,{{\it \_R}}^{5}cd+8\,{{\it \_R}}^{3}a{e}^{2}-4\,{{\it \_R}}^{3}bde+3\,{{\it \_R}}^{3}c{d}^{2}+{\it \_R}\,b{d}^{2}e-{\it \_R}\,c{d}^{3}}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e}-{\it \_R} \right ) }}-{\frac{bde}{2\,{c}^{2}} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-1}}+{\frac{5\,{d}^{2}}{8\,c} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-1}}+{\frac{{d}^{3}}{24\,c} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x)

[Out]

-1/6/c*e^(3/2)*x^3+1/8/c*e*(e*x^2+d)^(1/2)*x^2-3/4/c*e^(1/2)*x*d+1/24*(e*x^2+d)^
(3/2)/c+1/2/c^2*e^(3/2)*x*b-1/2/c^2*(e*x^2+d)^(1/2)*b*e+5/8/c*(e*x^2+d)^(1/2)*d+
1/4/c^2*sum(((-a*c*e^2+b^2*e^2-2*b*c*d*e+c^2*d^2)*_R^6+(4*a*b*e^3-5*a*c*d*e^2-3*
b^2*d*e^2+6*b*c*d^2*e-3*c^2*d^3)*_R^4+d*(-4*a*b*e^3+5*a*c*d*e^2+3*b^2*d*e^2-6*b*
c*d^2*e+3*c^2*d^3)*_R^2+a*c*d^3*e^2-b^2*d^3*e^2+2*b*c*d^4*e-c^2*d^5)/(_R^7*c+3*_
R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3*_R^3*c*d^2+_R*b*d^2*e-_R*c*d^3)*l
n((e*x^2+d)^(1/2)-x*e^(1/2)-_R),_R=RootOf(c*_Z^8+(4*b*e-4*c*d)*_Z^6+(16*a*e^2-8*
b*d*e+6*c*d^2)*_Z^4+(4*b*d^2*e-4*c*d^3)*_Z^2+c*d^4))-1/2/c^2*d/((e*x^2+d)^(1/2)-
x*e^(1/2))*b*e+5/8/c*d^2/((e*x^2+d)^(1/2)-x*e^(1/2))+1/24/c*d^3/((e*x^2+d)^(1/2)
-x*e^(1/2))^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} x^{3}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x^3/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^(3/2)*x^3/(c*x^4 + b*x^2 + a), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x^3/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (d + e x^{2}\right )^{\frac{3}{2}}}{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x**3*(d + e*x**2)**(3/2)/(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^(3/2)*x^3/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Timed out