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

Optimal. Leaf size=210 \[ \frac{a^{3/2} (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^4}-\frac{(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^4 d^{3/2}}+\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-10 a b c d+b^2 c^2\right )}{16 b^3 d}+\frac{x^3 \sqrt{c+d x^2} (7 b c-6 a d)}{24 b^2}+\frac{d x^5 \sqrt{c+d x^2}}{6 b} \]

[Out]

((b^2*c^2 - 10*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*b^3*d) + ((7*b*c - 6*
a*d)*x^3*Sqrt[c + d*x^2])/(24*b^2) + (d*x^5*Sqrt[c + d*x^2])/(6*b) + (a^(3/2)*(b
*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^4 - ((b
*c - 2*a*d)*(b^2*c^2 + 8*a*b*c*d - 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2
]])/(16*b^4*d^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.01949, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a^{3/2} (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^4}-\frac{(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^4 d^{3/2}}+\frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-10 a b c d+b^2 c^2\right )}{16 b^3 d}+\frac{x^3 \sqrt{c+d x^2} (7 b c-6 a d)}{24 b^2}+\frac{d x^5 \sqrt{c+d x^2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*c^2 - 10*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*b^3*d) + ((7*b*c - 6*
a*d)*x^3*Sqrt[c + d*x^2])/(24*b^2) + (d*x^5*Sqrt[c + d*x^2])/(6*b) + (a^(3/2)*(b
*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^4 - ((b
*c - 2*a*d)*(b^2*c^2 + 8*a*b*c*d - 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2
]])/(16*b^4*d^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 135.424, size = 197, normalized size = 0.94 \[ \frac{a^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b^{4}} + \frac{d x^{5} \sqrt{c + d x^{2}}}{6 b} - \frac{x^{3} \sqrt{c + d x^{2}} \left (6 a d - 7 b c\right )}{24 b^{2}} + \frac{x \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} - 10 a b c d + b^{2} c^{2}\right )}{16 b^{3} d} - \frac{\left (2 a d - b c\right ) \left (8 a^{2} d^{2} - 8 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 b^{4} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*(a*d - b*c)**(3/2)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/
b**4 + d*x**5*sqrt(c + d*x**2)/(6*b) - x**3*sqrt(c + d*x**2)*(6*a*d - 7*b*c)/(24
*b**2) + x*sqrt(c + d*x**2)*(8*a**2*d**2 - 10*a*b*c*d + b**2*c**2)/(16*b**3*d) -
(2*a*d - b*c)*(8*a**2*d**2 - 8*a*b*c*d - b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*
x**2))/(16*b**4*d**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.380343, size = 196, normalized size = 0.93 \[ \frac{48 a^{3/2} d^{3/2} (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )+b \sqrt{d} x \sqrt{c+d x^2} \left (24 a^2 d^2-6 a b d \left (5 c+2 d x^2\right )+b^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )-3 \left (16 a^3 d^3-24 a^2 b c d^2+6 a b^2 c^2 d+b^3 c^3\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{48 b^4 d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[d]*x*Sqrt[c + d*x^2]*(24*a^2*d^2 - 6*a*b*d*(5*c + 2*d*x^2) + b^2*(3*c^2
+ 14*c*d*x^2 + 8*d^2*x^4)) + 48*a^(3/2)*d^(3/2)*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b
*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] - 3*(b^3*c^3 + 6*a*b^2*c^2*d - 24*a^2*b*
c*d^2 + 16*a^3*d^3)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(48*b^4*d^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 2081, normalized size = 9.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/b^3*a^3/(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a
*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d+1/2/b^2*a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln
((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*
((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(
1/2))/(x+1/b*(-a*b)^(1/2)))*c^2+1/2/b^4*a^4/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln
((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*
((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(
1/2))/(x+1/b*(-a*b)^(1/2)))*d^2-1/2/b^4*a^4/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln
((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*
((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(
1/2))/(x-1/b*(-a*b)^(1/2)))*d^2-1/2/b^2*a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln
((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*
((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(
1/2))/(x-1/b*(-a*b)^(1/2)))*c^2+3/4/b^3*a^2*d^(1/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b
*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-
a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c-1/2/b^2*a^2/(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))
^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c+1/4/b^3*a^2*d*
((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(
1/2)*x+3/4/b^3*a^2*d^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+
((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(
1/2))*c-1/2/b^3*a^3/(-a*b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x
-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d+1/2/b^2*a^2/(-a*b)^(1/2)*((x-1/b*(-a*b)^
(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c-3/8/b^2*
a*c^2/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/24/b*c/d*x*(d*x^2+c)^(3/2)-1/16/b*
c^2/d*x*(d*x^2+c)^(1/2)-3/8/b^2*a*c*x*(d*x^2+c)^(1/2)+1/4/b^3*a^2*d*((x+1/b*(-a*
b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-1/6/b
^2*a^2/(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(
1/2))-(a*d-b*c)/b)^(3/2)-1/2/b^4*a^3*d^(3/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)
^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1
/2))-(a*d-b*c)/b)^(1/2))+1/6/b^2*a^2/(-a*b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*
(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-1/2/b^4*a^3*d^(3/2)*ln((d
*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-
a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+1/6/b*x*(d*x^2+c)^(5/2)/d-
1/16/b*c^3/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/4/b^2*a*x*(d*x^2+c)^(3/2)-1/b
^3*a^3/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(
x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^
(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d*c+1/b^3
*a^3/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-
1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d*c

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.46672, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/96*(24*(a*b*c*d - a^2*d^2)*sqrt(-a*b*c + a^2*d)*sqrt(d)*log(((b^2*c^2 - 8*a*
b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*((b*c - 2*a
*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^
2)) - 2*(8*b^3*d^2*x^5 + 2*(7*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(b^3*c^2 - 10*a*b^2
*c*d + 8*a^2*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) - 3*(b^3*c^3 + 6*a*b^2*c^2*d - 24
*a^2*b*c*d^2 + 16*a^3*d^3)*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/(
b^4*d^(3/2)), -1/48*(12*(a*b*c*d - a^2*d^2)*sqrt(-a*b*c + a^2*d)*sqrt(-d)*log(((
b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 -
4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 +
2*a*b*x^2 + a^2)) - (8*b^3*d^2*x^5 + 2*(7*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(b^3*c^
2 - 10*a*b^2*c*d + 8*a^2*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d) + 3*(b^3*c^3 + 6*a*b
^2*c^2*d - 24*a^2*b*c*d^2 + 16*a^3*d^3)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^4
*sqrt(-d)*d), -1/96*(48*(a*b*c*d - a^2*d^2)*sqrt(a*b*c - a^2*d)*sqrt(d)*arctan(-
1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)) - 2*(8*b^
3*d^2*x^5 + 2*(7*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(b^3*c^2 - 10*a*b^2*c*d + 8*a^2*
b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) - 3*(b^3*c^3 + 6*a*b^2*c^2*d - 24*a^2*b*c*d^2
+ 16*a^3*d^3)*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/(b^4*d^(3/2)),
-1/48*(24*(a*b*c*d - a^2*d^2)*sqrt(a*b*c - a^2*d)*sqrt(-d)*arctan(-1/2*((b*c -
2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)) - (8*b^3*d^2*x^5 + 2*
(7*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(b^3*c^2 - 10*a*b^2*c*d + 8*a^2*b*d^2)*x)*sqrt
(d*x^2 + c)*sqrt(-d) + 3*(b^3*c^3 + 6*a*b^2*c^2*d - 24*a^2*b*c*d^2 + 16*a^3*d^3)
*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^4*sqrt(-d)*d)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.257921, size = 358, normalized size = 1.7 \[ \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, d x^{2}}{b} + \frac{7 \, b^{9} c d^{4} - 6 \, a b^{8} d^{5}}{b^{10} d^{4}}\right )} x^{2} + \frac{3 \,{\left (b^{9} c^{2} d^{3} - 10 \, a b^{8} c d^{4} + 8 \, a^{2} b^{7} d^{5}\right )}}{b^{10} d^{4}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (a^{2} b^{2} c^{2} \sqrt{d} - 2 \, a^{3} b c d^{\frac{3}{2}} + a^{4} d^{\frac{5}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{4}} + \frac{{\left (b^{3} c^{3} \sqrt{d} + 6 \, a b^{2} c^{2} d^{\frac{3}{2}} - 24 \, a^{2} b c d^{\frac{5}{2}} + 16 \, a^{3} d^{\frac{7}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{32 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*d*x^2/b + (7*b^9*c*d^4 - 6*a*b^8*d^5)/(b^10*d^4))*x^2 + 3*(b^9*c^2*d^
3 - 10*a*b^8*c*d^4 + 8*a^2*b^7*d^5)/(b^10*d^4))*sqrt(d*x^2 + c)*x - (a^2*b^2*c^2
*sqrt(d) - 2*a^3*b*c*d^(3/2) + a^4*d^(5/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2
+ c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*b^4)
+ 1/32*(b^3*c^3*sqrt(d) + 6*a*b^2*c^2*d^(3/2) - 24*a^2*b*c*d^(5/2) + 16*a^3*d^(7
/2))*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b^4*d^2)