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

Optimal. Leaf size=157 \[ \frac{\sqrt{b} \left (15 a^2 d^2-20 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 d^3}-\frac{(b c-a d)^{5/2} \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d^3}-\frac{b x \sqrt{a+b x^2} (4 b c-7 a d)}{8 d^2}+\frac{b x \left (a+b x^2\right )^{3/2}}{4 d} \]

[Out]

-(b*(4*b*c - 7*a*d)*x*Sqrt[a + b*x^2])/(8*d^2) + (b*x*(a + b*x^2)^(3/2))/(4*d) +
(Sqrt[b]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(8*d^3) - ((b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a +
b*x^2])])/(Sqrt[c]*d^3)

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Rubi [A]  time = 0.482411, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{b} \left (15 a^2 d^2-20 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 d^3}-\frac{(b c-a d)^{5/2} \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d^3}-\frac{b x \sqrt{a+b x^2} (4 b c-7 a d)}{8 d^2}+\frac{b x \left (a+b x^2\right )^{3/2}}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(4*b*c - 7*a*d)*x*Sqrt[a + b*x^2])/(8*d^2) + (b*x*(a + b*x^2)^(3/2))/(4*d) +
(Sqrt[b]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(8*d^3) - ((b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a +
b*x^2])])/(Sqrt[c]*d^3)

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Rubi in Sympy [A]  time = 69.3886, size = 146, normalized size = 0.93 \[ \frac{\sqrt{b} \left (15 a^{2} d^{2} - 20 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 d^{3}} + \frac{b x \left (a + b x^{2}\right )^{\frac{3}{2}}}{4 d} + \frac{b x \sqrt{a + b x^{2}} \left (7 a d - 4 b c\right )}{8 d^{2}} + \frac{\left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{\sqrt{c} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*(15*a**2*d**2 - 20*a*b*c*d + 8*b**2*c**2)*atanh(sqrt(b)*x/sqrt(a + b*x**
2))/(8*d**3) + b*x*(a + b*x**2)**(3/2)/(4*d) + b*x*sqrt(a + b*x**2)*(7*a*d - 4*b
*c)/(8*d**2) + (a*d - b*c)**(5/2)*atan(x*sqrt(a*d - b*c)/(sqrt(c)*sqrt(a + b*x**
2)))/(sqrt(c)*d**3)

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Mathematica [A]  time = 0.18899, size = 140, normalized size = 0.89 \[ \frac{\sqrt{b} \left (15 a^2 d^2-20 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+b d x \sqrt{a+b x^2} \left (9 a d-4 b c+2 b d x^2\right )+\frac{8 (a d-b c)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c}}}{8 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b*d*x*Sqrt[a + b*x^2]*(-4*b*c + 9*a*d + 2*b*d*x^2) + (8*(-(b*c) + a*d)^(5/2)*Ar
cTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/Sqrt[c] + Sqrt[b]*(8*b^2
*c^2 - 20*a*b*c*d + 15*a^2*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*d^3)

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Maple [B]  time = 0.027, size = 3053, normalized size = 19.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32431, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*((8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2
+ a)*sqrt(b)*x - a) + 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt((b*c - a*d)/c)*log
(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^
2 - 4*(a*c^2*x + (2*b*c^2 - a*c*d)*x^3)*sqrt(b*x^2 + a)*sqrt((b*c - a*d)/c))/(d^
2*x^4 + 2*c*d*x^2 + c^2)) + 2*(2*b^2*d^2*x^3 - (4*b^2*c*d - 9*a*b*d^2)*x)*sqrt(b
*x^2 + a))/d^3, 1/8*((8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*sqrt(-b)*arctan(b*x/(
sqrt(b*x^2 + a)*sqrt(-b))) + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt((b*c - a*d)/
c)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c
*d)*x^2 - 4*(a*c^2*x + (2*b*c^2 - a*c*d)*x^3)*sqrt(b*x^2 + a)*sqrt((b*c - a*d)/c
))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + (2*b^2*d^2*x^3 - (4*b^2*c*d - 9*a*b*d^2)*x)*sq
rt(b*x^2 + a))/d^3, 1/16*(8*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/c)
*arctan(-1/2*((2*b*c - a*d)*x^2 + a*c)/(sqrt(b*x^2 + a)*c*x*sqrt(-(b*c - a*d)/c)
)) + (8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 +
a)*sqrt(b)*x - a) + 2*(2*b^2*d^2*x^3 - (4*b^2*c*d - 9*a*b*d^2)*x)*sqrt(b*x^2 +
a))/d^3, 1/8*((8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*sqrt(-b)*arctan(b*x/(sqrt(b*
x^2 + a)*sqrt(-b))) + 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/c)*arc
tan(-1/2*((2*b*c - a*d)*x^2 + a*c)/(sqrt(b*x^2 + a)*c*x*sqrt(-(b*c - a*d)/c))) +
(2*b^2*d^2*x^3 - (4*b^2*c*d - 9*a*b*d^2)*x)*sqrt(b*x^2 + a))/d^3]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.408494, size = 290, normalized size = 1.85 \[ \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (\frac{2 \, b^{2} x^{2}}{d} - \frac{4 \, b^{4} c d^{4} - 9 \, a b^{3} d^{5}}{b^{2} d^{6}}\right )} x - \frac{{\left (8 \, b^{\frac{5}{2}} c^{2} - 20 \, a b^{\frac{3}{2}} c d + 15 \, a^{2} \sqrt{b} d^{2}\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{16 \, d^{3}} + \frac{{\left (b^{\frac{7}{2}} c^{3} - 3 \, a b^{\frac{5}{2}} c^{2} d + 3 \, a^{2} b^{\frac{3}{2}} c d^{2} - a^{3} \sqrt{b} d^{3}\right )} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{\sqrt{-b^{2} c^{2} + a b c d} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(b*x^2 + a)*(2*b^2*x^2/d - (4*b^4*c*d^4 - 9*a*b^3*d^5)/(b^2*d^6))*x - 1/
16*(8*b^(5/2)*c^2 - 20*a*b^(3/2)*c*d + 15*a^2*sqrt(b)*d^2)*ln((sqrt(b)*x - sqrt(
b*x^2 + a))^2)/d^3 + (b^(7/2)*c^3 - 3*a*b^(5/2)*c^2*d + 3*a^2*b^(3/2)*c*d^2 - a^
3*sqrt(b)*d^3)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt
(-b^2*c^2 + a*b*c*d))/(sqrt(-b^2*c^2 + a*b*c*d)*d^3)