3.346 \(\int \frac{x^7 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=406 \[ -\frac{\left (-\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{\left (d+e x^2\right )^{3/2} (b e+c d)}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2} \]

[Out]

((b^2 - a*c)*Sqrt[d + e*x^2])/c^3 - ((c*d + b*e)*(d + e*x^2)^(3/2))/(3*c^2*e^2)
+ (d + e*x^2)^(5/2)/(5*c*e^2) - ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e - (b^3*c
*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh
[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sq
rt[2]*c^(7/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b^2*c*d - a*c^2*d - b
^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/S
qrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(7/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

_______________________________________________________________________________________

Rubi [A]  time = 17.8635, antiderivative size = 406, 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 \[ -\frac{\left (-\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{\left (d+e x^2\right )^{3/2} (b e+c d)}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]

[Out]

((b^2 - a*c)*Sqrt[d + e*x^2])/c^3 - ((c*d + b*e)*(d + e*x^2)^(3/2))/(3*c^2*e^2)
+ (d + e*x^2)^(5/2)/(5*c*e^2) - ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e - (b^3*c
*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh
[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sq
rt[2]*c^(7/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((b^2*c*d - a*c^2*d - b
^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/S
qrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(7/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

_______________________________________________________________________________________

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**7*(e*x**2+d)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.51953, size = 476, normalized size = 1.17 \[ \frac{\sqrt{d+e x^2} \left (-5 c e \left (3 a e+b \left (d+e x^2\right )\right )+15 b^2 e^2+c^2 \left (-2 d^2+d e x^2+3 e^2 x^4\right )\right )}{15 c^3 e^2}+\frac{\left (a c^2 \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^2 c \left (4 a e-d \sqrt{b^2-4 a c}\right )-a b c \left (2 e \sqrt{b^2-4 a c}+3 c d\right )+b^3 \left (e \sqrt{b^2-4 a c}+c d\right )+b^4 (-e)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\left (a c^2 \left (d \sqrt{b^2-4 a c}+2 a e\right )-b^2 c \left (d \sqrt{b^2-4 a c}+4 a e\right )+a b c \left (3 c d-2 e \sqrt{b^2-4 a c}\right )+b^3 \left (e \sqrt{b^2-4 a c}-c d\right )+b^4 e\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*Sqrt[d + e*x^2])/(a + b*x^2 + c*x^4),x]

[Out]

(Sqrt[d + e*x^2]*(15*b^2*e^2 + c^2*(-2*d^2 + d*e*x^2 + 3*e^2*x^4) - 5*c*e*(3*a*e
+ b*(d + e*x^2))))/(15*c^3*e^2) + ((-(b^4*e) + a*c^2*(Sqrt[b^2 - 4*a*c]*d - 2*a
*e) + b^2*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^3*(c*d + Sqrt[b^2 - 4*a*c]*e) -
a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2]
)/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*S
qrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + ((b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d
+ 2*a*e) - b^2*c*(Sqrt[b^2 - 4*a*c]*d + 4*a*e) + a*b*c*(3*c*d - 2*Sqrt[b^2 - 4*a
*c]*e) + b^3*(-(c*d) + Sqrt[b^2 - 4*a*c]*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^(7/2)*Sqrt[b^2 - 4*a
*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

_______________________________________________________________________________________

Maple [C]  time = 0.084, size = 496, normalized size = 1.2 \[{\frac{{x}^{2}}{5\,ce} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,d}{15\,{e}^{2}c} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{b}{3\,{c}^{2}e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{ax}{2\,{c}^{2}}\sqrt{e}}-{\frac{x{b}^{2}}{2\,{c}^{3}}\sqrt{e}}-{\frac{a}{2\,{c}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{{b}^{2}}{2\,{c}^{3}}\sqrt{e{x}^{2}+d}}-{\frac{1}{4\,{c}^{3}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{6}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{4}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){{\it \_Z}}^{2}+c{d}^{4} \right ) }{\frac{ \left ( -2\,abce+a{c}^{2}d+{b}^{3}e-{b}^{2}cd \right ){{\it \_R}}^{6}+ \left ( -4\,{a}^{2}c{e}^{2}+4\,a{b}^{2}{e}^{2}+2\,abcde-3\,a{c}^{2}{d}^{2}-3\,{b}^{3}de+3\,{b}^{2}c{d}^{2} \right ){{\it \_R}}^{4}+d \left ( 4\,{a}^{2}c{e}^{2}-4\,a{b}^{2}{e}^{2}-2\,abcde+3\,a{c}^{2}{d}^{2}+3\,{b}^{3}de-3\,{b}^{2}c{d}^{2} \right ){{\it \_R}}^{2}+2\,abc{d}^{3}e-a{c}^{2}{d}^{4}-{b}^{3}{d}^{3}e+{b}^{2}c{d}^{4}}{{{\it \_R}}^{7}c+3\,{{\it \_R}}^{5}be-3\,{{\it \_R}}^{5}cd+8\,{{\it \_R}}^{3}a{e}^{2}-4\,{{\it \_R}}^{3}bde+3\,{{\it \_R}}^{3}c{d}^{2}+{\it \_R}\,b{d}^{2}e-{\it \_R}\,c{d}^{3}}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e}-{\it \_R} \right ) }}-{\frac{ad}{2\,{c}^{2}} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-1}}+{\frac{{b}^{2}d}{2\,{c}^{3}} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a),x)

[Out]

1/5/c*x^2*(e*x^2+d)^(3/2)/e-2/15/c*d/e^2*(e*x^2+d)^(3/2)-1/3/c^2*b*(e*x^2+d)^(3/
2)/e+1/2/c^2*e^(1/2)*x*a-1/2/c^3*e^(1/2)*x*b^2-1/2/c^2*(e*x^2+d)^(1/2)*a+1/2/c^3
*(e*x^2+d)^(1/2)*b^2-1/4/c^3*sum(((-2*a*b*c*e+a*c^2*d+b^3*e-b^2*c*d)*_R^6+(-4*a^
2*c*e^2+4*a*b^2*e^2+2*a*b*c*d*e-3*a*c^2*d^2-3*b^3*d*e+3*b^2*c*d^2)*_R^4+d*(4*a^2
*c*e^2-4*a*b^2*e^2-2*a*b*c*d*e+3*a*c^2*d^2+3*b^3*d*e-3*b^2*c*d^2)*_R^2+2*a*b*c*d
^3*e-a*c^2*d^4-b^3*d^3*e+b^2*c*d^4)/(_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)*ln((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))*a+1/2/c^3*d/((e*x^2+d)
^(1/2)-x*e^(1/2))*b^2

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x^{2} + d} x^{7}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x^2 + d)*x^7/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x^2 + d)*x^7/(c*x^4 + b*x^2 + a), x)

_______________________________________________________________________________________

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(sqrt(e*x^2 + d)*x^7/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7} \sqrt{d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(e*x**2+d)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(x**7*sqrt(d + e*x**2)/(a + b*x**2 + c*x**4), x)

_______________________________________________________________________________________

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(sqrt(e*x^2 + d)*x^7/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Timed out