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

Optimal. Leaf size=195 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^4}+\frac{(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} b^4}+\frac{d x \sqrt{c+d x^2} (11 b c-12 a d)}{8 b^3}-\frac{x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{3 d x \left (c+d x^2\right )^{3/2}}{4 b^2} \]

[Out]

(d*(11*b*c - 12*a*d)*x*Sqrt[c + d*x^2])/(8*b^3) + (3*d*x*(c + d*x^2)^(3/2))/(4*b
^2) - (x*(c + d*x^2)^(5/2))/(2*b*(a + b*x^2)) + ((b*c - 6*a*d)*(b*c - a*d)^(3/2)
*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*b^4) + (Sqrt[
d]*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/
(8*b^4)

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Rubi [A]  time = 0.634916, antiderivative size = 195, 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 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^4}+\frac{(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} b^4}+\frac{d x \sqrt{c+d x^2} (11 b c-12 a d)}{8 b^3}-\frac{x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{3 d x \left (c+d x^2\right )^{3/2}}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(11*b*c - 12*a*d)*x*Sqrt[c + d*x^2])/(8*b^3) + (3*d*x*(c + d*x^2)^(3/2))/(4*b
^2) - (x*(c + d*x^2)^(5/2))/(2*b*(a + b*x^2)) + ((b*c - 6*a*d)*(b*c - a*d)^(3/2)
*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*b^4) + (Sqrt[
d]*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/
(8*b^4)

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Rubi in Sympy [A]  time = 94.1015, size = 182, normalized size = 0.93 \[ - \frac{x \left (c + d x^{2}\right )^{\frac{5}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{3 d x \left (c + d x^{2}\right )^{\frac{3}{2}}}{4 b^{2}} - \frac{d x \sqrt{c + d x^{2}} \left (12 a d - 11 b c\right )}{8 b^{3}} + \frac{\sqrt{d} \left (24 a^{2} d^{2} - 40 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 b^{4}} - \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (6 a d - b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \sqrt{a} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(c + d*x**2)**(5/2)/(2*b*(a + b*x**2)) + 3*d*x*(c + d*x**2)**(3/2)/(4*b**2) -
d*x*sqrt(c + d*x**2)*(12*a*d - 11*b*c)/(8*b**3) + sqrt(d)*(24*a**2*d**2 - 40*a*
b*c*d + 15*b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(8*b**4) - (a*d - b*c)**
(3/2)*(6*a*d - b*c)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(2*sqrt(
a)*b**4)

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Mathematica [A]  time = 0.374174, size = 173, normalized size = 0.89 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+b x \sqrt{c+d x^2} \left (-\frac{4 (b c-a d)^2}{a+b x^2}+d (9 b c-8 a d)+2 b d^2 x^2\right )+\frac{4 (b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a}}}{8 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*Sqrt[c + d*x^2]*(d*(9*b*c - 8*a*d) + 2*b*d^2*x^2 - (4*(b*c - a*d)^2)/(a + b
*x^2)) + (4*(b*c - 6*a*d)*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*
Sqrt[c + d*x^2])])/Sqrt[a] + Sqrt[d]*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*Log[
d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*b^4)

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Maple [B]  time = 0.026, size = 7459, normalized size = 38.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.42925, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*((15*a*b^2*c^2 - 40*a^2*b*c*d + 24*a^3*d^2 + (15*b^3*c^2 - 40*a*b^2*c*d +
24*a^2*b*d^2)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(
a*b^2*c^2 - 7*a^2*b*c*d + 6*a^3*d^2 + (b^3*c^2 - 7*a*b^2*c*d + 6*a^2*b*d^2)*x^2)
*sqrt(-(b*c - a*d)/a)*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*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c)
*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(2*b^3*d^2*x^5 + 3*(3*b^
3*c*d - 2*a*b^2*d^2)*x^3 - (4*b^3*c^2 - 17*a*b^2*c*d + 12*a^2*b*d^2)*x)*sqrt(d*x
^2 + c))/(b^5*x^2 + a*b^4), 1/8*((15*a*b^2*c^2 - 40*a^2*b*c*d + 24*a^3*d^2 + (15
*b^3*c^2 - 40*a*b^2*c*d + 24*a^2*b*d^2)*x^2)*sqrt(-d)*arctan(d*x/(sqrt(d*x^2 + c
)*sqrt(-d))) + (a*b^2*c^2 - 7*a^2*b*c*d + 6*a^3*d^2 + (b^3*c^2 - 7*a*b^2*c*d + 6
*a^2*b*d^2)*x^2)*sqrt(-(b*c - a*d)/a)*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*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)
*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + (2*b^3*d^2
*x^5 + 3*(3*b^3*c*d - 2*a*b^2*d^2)*x^3 - (4*b^3*c^2 - 17*a*b^2*c*d + 12*a^2*b*d^
2)*x)*sqrt(d*x^2 + c))/(b^5*x^2 + a*b^4), -1/16*(4*(a*b^2*c^2 - 7*a^2*b*c*d + 6*
a^3*d^2 + (b^3*c^2 - 7*a*b^2*c*d + 6*a^2*b*d^2)*x^2)*sqrt((b*c - a*d)/a)*arctan(
-1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(d*x^2 + c)*a*x*sqrt((b*c - a*d)/a))) - (15*
a*b^2*c^2 - 40*a^2*b*c*d + 24*a^3*d^2 + (15*b^3*c^2 - 40*a*b^2*c*d + 24*a^2*b*d^
2)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 2*(2*b^3*d^2*x
^5 + 3*(3*b^3*c*d - 2*a*b^2*d^2)*x^3 - (4*b^3*c^2 - 17*a*b^2*c*d + 12*a^2*b*d^2)
*x)*sqrt(d*x^2 + c))/(b^5*x^2 + a*b^4), 1/8*((15*a*b^2*c^2 - 40*a^2*b*c*d + 24*a
^3*d^2 + (15*b^3*c^2 - 40*a*b^2*c*d + 24*a^2*b*d^2)*x^2)*sqrt(-d)*arctan(d*x/(sq
rt(d*x^2 + c)*sqrt(-d))) - 2*(a*b^2*c^2 - 7*a^2*b*c*d + 6*a^3*d^2 + (b^3*c^2 - 7
*a*b^2*c*d + 6*a^2*b*d^2)*x^2)*sqrt((b*c - a*d)/a)*arctan(-1/2*((b*c - 2*a*d)*x^
2 - a*c)/(sqrt(d*x^2 + c)*a*x*sqrt((b*c - a*d)/a))) + (2*b^3*d^2*x^5 + 3*(3*b^3*
c*d - 2*a*b^2*d^2)*x^3 - (4*b^3*c^2 - 17*a*b^2*c*d + 12*a^2*b*d^2)*x)*sqrt(d*x^2
+ c))/(b^5*x^2 + a*b^4)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.557498, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x