3.694 \(\int \frac{x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx\)

Optimal. Leaf size=291 \[ \frac{a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^5}+\frac{x^3 \sqrt{c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{192 b^3}+\frac{x \sqrt{c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{128 b^4 d}-\frac{\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 b^5 d^{3/2}}+\frac{d x^5 \sqrt{c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b} \]

[Out]

((5*b^3*c^3 - 88*a*b^2*c^2*d + 144*a^2*b*c*d^2 - 64*a^3*d^3)*x*Sqrt[c + d*x^2])/
(128*b^4*d) + ((59*b^2*c^2 - 104*a*b*c*d + 48*a^2*d^2)*x^3*Sqrt[c + d*x^2])/(192
*b^3) + (d*(11*b*c - 8*a*d)*x^5*Sqrt[c + d*x^2])/(48*b^2) + (d*x^5*(c + d*x^2)^(
3/2))/(8*b) + (a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqr
t[c + d*x^2])])/b^5 - ((5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a
^3*b*c*d^3 - 128*a^4*d^4)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*b^5*d^(3/2)
)

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Rubi [A]  time = 1.43642, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^5}+\frac{x^3 \sqrt{c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{192 b^3}+\frac{x \sqrt{c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{128 b^4 d}-\frac{\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 b^5 d^{3/2}}+\frac{d x^5 \sqrt{c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2),x]

[Out]

((5*b^3*c^3 - 88*a*b^2*c^2*d + 144*a^2*b*c*d^2 - 64*a^3*d^3)*x*Sqrt[c + d*x^2])/
(128*b^4*d) + ((59*b^2*c^2 - 104*a*b*c*d + 48*a^2*d^2)*x^3*Sqrt[c + d*x^2])/(192
*b^3) + (d*(11*b*c - 8*a*d)*x^5*Sqrt[c + d*x^2])/(48*b^2) + (d*x^5*(c + d*x^2)^(
3/2))/(8*b) + (a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqr
t[c + d*x^2])])/b^5 - ((5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a
^3*b*c*d^3 - 128*a^4*d^4)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*b^5*d^(3/2)
)

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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**4*(d*x**2+c)**(5/2)/(b*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.411166, size = 247, normalized size = 0.85 \[ \frac{384 a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )+\frac{b x \sqrt{c+d x^2} \left (-192 a^3 d^3+48 a^2 b d^2 \left (9 c+2 d x^2\right )-8 a b^2 d \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+b^3 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )\right )}{d}+\frac{3 \left (128 a^4 d^4-320 a^3 b c d^3+240 a^2 b^2 c^2 d^2-40 a b^3 c^3 d-5 b^4 c^4\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{d^{3/2}}}{384 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2),x]

[Out]

((b*x*Sqrt[c + d*x^2]*(-192*a^3*d^3 + 48*a^2*b*d^2*(9*c + 2*d*x^2) - 8*a*b^2*d*(
33*c^2 + 26*c*d*x^2 + 8*d^2*x^4) + b^3*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 +
48*d^3*x^6)))/d + 384*a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqr
t[a]*Sqrt[c + d*x^2])] + (3*(-5*b^4*c^4 - 40*a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 -
320*a^3*b*c*d^3 + 128*a^4*d^4)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/d^(3/2))/(38
4*b^5)

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Maple [B]  time = 0.032, size = 3373, normalized size = 11.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(d*x^2+c)^(5/2)/(b*x^2+a),x)

[Out]

-1/6/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)^(3/2)*d+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)*c-1/4/b^4*a^3*d^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-5/16/b^2*a*c^3/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/8/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)^(3/2)*
x-5/4/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))
*c+1/2/b^4*a^4/(-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^2+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^2-1/48/b*c/
d*x*(d*x^2+c)^(5/2)-5/24/b^2*a*c*x*(d*x^2+c)^(3/2)+15/16/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^2+1/6/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)
^(3/2)*d-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)*c-1/4/b^4*a^3*d^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-5/4/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))*c-5/16/b^2*a*c^2
*x*(d*x^2+c)^(1/2)-1/2/b^4*a^4/(-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^2-1/2/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^2-5/128/b*c^3/d*x*(d*x^2+c)^(1/2)-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^3+1/8/b*x*(d*x^2+c)^(7/2)/d-5/128/b*c^4/d^(
3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/2/b^5*a^4*d^(5/2)*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))+7/16/b^3*a^2*d*c*((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^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*c+7/16/b^3*a^2*d*c*((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^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*c-1/2/b^5*a^5/(-
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^3+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^3+1/2/b^5*a^5/(-
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^3-5/192/b*c^2/d*
x*(d*x^2+c)^(3/2)-1/6/b^2*a*x*(d*x^2+c)^(5/2)+1/8/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)^(3/2)*x+15/16/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^2-1/10/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)^(5/2)+1/2/b^5*a^4*d^(5/2)*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))+1/10/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)^(5/2)-3/2/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*c+3/2/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^2+3/2/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*c-3/2/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^2

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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)^(5/2)*x^4/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 9.06747, 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)^(5/2)*x^4/(b*x^2 + a),x, algorithm="fricas")

[Out]

[1/768*(192*(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*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*(48*b^4*d^3*x^7 + 8*(17*b^4*c*d^2 - 8*a*b^3*d^3)*x^
5 + 2*(59*b^4*c^2*d - 104*a*b^3*c*d^2 + 48*a^2*b^2*d^3)*x^3 + 3*(5*b^4*c^3 - 88*
a*b^3*c^2*d + 144*a^2*b^2*c*d^2 - 64*a^3*b*d^3)*x)*sqrt(d*x^2 + c)*sqrt(d) - 3*(
5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4
)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/(b^5*d^(3/2)), 1/384*(96*
(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*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)) + (48*b^4*d^3*x^7 + 8*(17*b^4*c*d^2 - 8*a*b^3*d^3)*x^5 + 2*(59*b^4
*c^2*d - 104*a*b^3*c*d^2 + 48*a^2*b^2*d^3)*x^3 + 3*(5*b^4*c^3 - 88*a*b^3*c^2*d +
144*a^2*b^2*c*d^2 - 64*a^3*b*d^3)*x)*sqrt(d*x^2 + c)*sqrt(-d) - 3*(5*b^4*c^4 +
40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*arctan(sqr
t(-d)*x/sqrt(d*x^2 + c)))/(b^5*sqrt(-d)*d), -1/768*(384*(a*b^2*c^2*d - 2*a^2*b*c
*d^2 + a^3*d^3)*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*(48*b^4*d^3*x^7 + 8*(17*b^4*c*d^2
- 8*a*b^3*d^3)*x^5 + 2*(59*b^4*c^2*d - 104*a*b^3*c*d^2 + 48*a^2*b^2*d^3)*x^3 +
3*(5*b^4*c^3 - 88*a*b^3*c^2*d + 144*a^2*b^2*c*d^2 - 64*a^3*b*d^3)*x)*sqrt(d*x^2
+ c)*sqrt(d) + 3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c
*d^3 - 128*a^4*d^4)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/(b^5*d^
(3/2)), -1/384*(192*(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*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)) - (48*b^4*d^3*x^7 + 8*(17*b^4*c*d^2 - 8*a*b^3*d^3)*x^5 + 2*(59*b^4*c^2*d
- 104*a*b^3*c*d^2 + 48*a^2*b^2*d^3)*x^3 + 3*(5*b^4*c^3 - 88*a*b^3*c^2*d + 144*a
^2*b^2*c*d^2 - 64*a^3*b*d^3)*x)*sqrt(d*x^2 + c)*sqrt(-d) + 3*(5*b^4*c^4 + 40*a*b
^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*arctan(sqrt(-d)*
x/sqrt(d*x^2 + c)))/(b^5*sqrt(-d)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (c + d x^{2}\right )^{\frac{5}{2}}}{a + b x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(d*x**2+c)**(5/2)/(b*x**2+a),x)

[Out]

Integral(x**4*(c + d*x**2)**(5/2)/(a + b*x**2), x)

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GIAC/XCAS [A]  time = 0.260082, size = 483, normalized size = 1.66 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (\frac{6 \, d^{2} x^{2}}{b} + \frac{17 \, b^{14} c d^{7} - 8 \, a b^{13} d^{8}}{b^{15} d^{6}}\right )} x^{2} + \frac{59 \, b^{14} c^{2} d^{6} - 104 \, a b^{13} c d^{7} + 48 \, a^{2} b^{12} d^{8}}{b^{15} d^{6}}\right )} x^{2} + \frac{3 \,{\left (5 \, b^{14} c^{3} d^{5} - 88 \, a b^{13} c^{2} d^{6} + 144 \, a^{2} b^{12} c d^{7} - 64 \, a^{3} b^{11} d^{8}\right )}}{b^{15} d^{6}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (a^{2} b^{3} c^{3} \sqrt{d} - 3 \, a^{3} b^{2} c^{2} d^{\frac{3}{2}} + 3 \, a^{4} b c d^{\frac{5}{2}} - a^{5} d^{\frac{7}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{5}} + \frac{{\left (5 \, b^{4} c^{4} \sqrt{d} + 40 \, a b^{3} c^{3} d^{\frac{3}{2}} - 240 \, a^{2} b^{2} c^{2} d^{\frac{5}{2}} + 320 \, a^{3} b c d^{\frac{7}{2}} - 128 \, a^{4} d^{\frac{9}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{256 \, b^{5} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)*x^4/(b*x^2 + a),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*d^2*x^2/b + (17*b^14*c*d^7 - 8*a*b^13*d^8)/(b^15*d^6))*x^2 + (59*
b^14*c^2*d^6 - 104*a*b^13*c*d^7 + 48*a^2*b^12*d^8)/(b^15*d^6))*x^2 + 3*(5*b^14*c
^3*d^5 - 88*a*b^13*c^2*d^6 + 144*a^2*b^12*c*d^7 - 64*a^3*b^11*d^8)/(b^15*d^6))*s
qrt(d*x^2 + c)*x - (a^2*b^3*c^3*sqrt(d) - 3*a^3*b^2*c^2*d^(3/2) + 3*a^4*b*c*d^(5
/2) - a^5*d^(7/2))*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^5) + 1/256*(5*b^4*c^4*sqrt(d
) + 40*a*b^3*c^3*d^(3/2) - 240*a^2*b^2*c^2*d^(5/2) + 320*a^3*b*c*d^(7/2) - 128*a
^4*d^(9/2))*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b^5*d^2)