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

Optimal. Leaf size=149 \[ \frac{5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 b c-a d)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 b c-a d)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 b c-a d)}{192 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b} \]

[Out]

(5*a^2*(8*b*c - a*d)*x*Sqrt[a + b*x^2])/(128*b) + (5*a*(8*b*c - a*d)*x*(a + b*x^
2)^(3/2))/(192*b) + ((8*b*c - a*d)*x*(a + b*x^2)^(5/2))/(48*b) + (d*x*(a + b*x^2
)^(7/2))/(8*b) + (5*a^3*(8*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128
*b^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.134588, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 a^3 (8 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{3/2}}+\frac{5 a^2 x \sqrt{a+b x^2} (8 b c-a d)}{128 b}+\frac{x \left (a+b x^2\right )^{5/2} (8 b c-a d)}{48 b}+\frac{5 a x \left (a+b x^2\right )^{3/2} (8 b c-a d)}{192 b}+\frac{d x \left (a+b x^2\right )^{7/2}}{8 b} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^2*(8*b*c - a*d)*x*Sqrt[a + b*x^2])/(128*b) + (5*a*(8*b*c - a*d)*x*(a + b*x^
2)^(3/2))/(192*b) + ((8*b*c - a*d)*x*(a + b*x^2)^(5/2))/(48*b) + (d*x*(a + b*x^2
)^(7/2))/(8*b) + (5*a^3*(8*b*c - a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128
*b^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.7114, size = 134, normalized size = 0.9 \[ - \frac{5 a^{3} \left (a d - 8 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{3}{2}}} - \frac{5 a^{2} x \sqrt{a + b x^{2}} \left (a d - 8 b c\right )}{128 b} - \frac{5 a x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - 8 b c\right )}{192 b} + \frac{d x \left (a + b x^{2}\right )^{\frac{7}{2}}}{8 b} - \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - 8 b c\right )}{48 b} \]

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]

-5*a**3*(a*d - 8*b*c)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(128*b**(3/2)) - 5*a**2*
x*sqrt(a + b*x**2)*(a*d - 8*b*c)/(128*b) - 5*a*x*(a + b*x**2)**(3/2)*(a*d - 8*b*
c)/(192*b) + d*x*(a + b*x**2)**(7/2)/(8*b) - x*(a + b*x**2)**(5/2)*(a*d - 8*b*c)
/(48*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.132819, size = 121, normalized size = 0.81 \[ \sqrt{a+b x^2} \left (\frac{a^2 x (5 a d+88 b c)}{128 b}+\frac{1}{48} b x^5 (17 a d+8 b c)+\frac{1}{192} a x^3 (59 a d+104 b c)+\frac{1}{8} b^2 d x^7\right )-\frac{5 a^3 (a d-8 b c) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{128 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a^2*(88*b*c + 5*a*d)*x)/(128*b) + (a*(104*b*c + 59*a*d)*x^3)/1
92 + (b*(8*b*c + 17*a*d)*x^5)/48 + (b^2*d*x^7)/8) - (5*a^3*(-8*b*c + a*d)*Log[b*
x + Sqrt[b]*Sqrt[a + b*x^2]])/(128*b^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 166, normalized size = 1.1 \[{\frac{cx}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,acx}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}cx}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{3}c}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{dx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{adx}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,d{a}^{2}x}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,d{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,d{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Fricas [A]  time = 0.325364, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, b^{3} d x^{7} + 8 \,{\left (8 \, b^{3} c + 17 \, a b^{2} d\right )} x^{5} + 2 \,{\left (104 \, a b^{2} c + 59 \, a^{2} b d\right )} x^{3} + 3 \,{\left (88 \, a^{2} b c + 5 \, a^{3} d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (8 \, a^{3} b c - a^{4} d\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{768 \, b^{\frac{3}{2}}}, \frac{{\left (48 \, b^{3} d x^{7} + 8 \,{\left (8 \, b^{3} c + 17 \, a b^{2} d\right )} x^{5} + 2 \,{\left (104 \, a b^{2} c + 59 \, a^{2} b d\right )} x^{3} + 3 \,{\left (88 \, a^{2} b c + 5 \, a^{3} d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (8 \, a^{3} b c - a^{4} d\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{384 \, \sqrt{-b} b}\right ] \]

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]

[1/768*(2*(48*b^3*d*x^7 + 8*(8*b^3*c + 17*a*b^2*d)*x^5 + 2*(104*a*b^2*c + 59*a^2
*b*d)*x^3 + 3*(88*a^2*b*c + 5*a^3*d)*x)*sqrt(b*x^2 + a)*sqrt(b) - 15*(8*a^3*b*c
- a^4*d)*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(3/2), 1/384*((48
*b^3*d*x^7 + 8*(8*b^3*c + 17*a*b^2*d)*x^5 + 2*(104*a*b^2*c + 59*a^2*b*d)*x^3 + 3
*(88*a^2*b*c + 5*a^3*d)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 15*(8*a^3*b*c - a^4*d)*arc
tan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b)]

_______________________________________________________________________________________

Sympy [A]  time = 88.5209, size = 316, normalized size = 2.12 \[ \frac{5 a^{\frac{7}{2}} d x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{5}{2}} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} c x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 a^{\frac{5}{2}} d x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} b c x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 a^{\frac{3}{2}} b d x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 \sqrt{a} b^{2} c x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{2} d x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 a^{4} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{5 a^{3} c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{b^{3} c x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} d x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(7/2)*d*x/(128*b*sqrt(1 + b*x**2/a)) + a**(5/2)*c*x*sqrt(1 + b*x**2/a)/2 +
3*a**(5/2)*c*x/(16*sqrt(1 + b*x**2/a)) + 133*a**(5/2)*d*x**3/(384*sqrt(1 + b*x**
2/a)) + 35*a**(3/2)*b*c*x**3/(48*sqrt(1 + b*x**2/a)) + 127*a**(3/2)*b*d*x**5/(19
2*sqrt(1 + b*x**2/a)) + 17*sqrt(a)*b**2*c*x**5/(24*sqrt(1 + b*x**2/a)) + 23*sqrt
(a)*b**2*d*x**7/(48*sqrt(1 + b*x**2/a)) - 5*a**4*d*asinh(sqrt(b)*x/sqrt(a))/(128
*b**(3/2)) + 5*a**3*c*asinh(sqrt(b)*x/sqrt(a))/(16*sqrt(b)) + b**3*c*x**7/(6*sqr
t(a)*sqrt(1 + b*x**2/a)) + b**3*d*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.337106, size = 182, normalized size = 1.22 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, b^{2} d x^{2} + \frac{8 \, b^{8} c + 17 \, a b^{7} d}{b^{6}}\right )} x^{2} + \frac{104 \, a b^{7} c + 59 \, a^{2} b^{6} d}{b^{6}}\right )} x^{2} + \frac{3 \,{\left (88 \, a^{2} b^{6} c + 5 \, a^{3} b^{5} d\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x - \frac{5 \,{\left (8 \, a^{3} b c - a^{4} d\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{3}{2}}} \]

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]

1/384*(2*(4*(6*b^2*d*x^2 + (8*b^8*c + 17*a*b^7*d)/b^6)*x^2 + (104*a*b^7*c + 59*a
^2*b^6*d)/b^6)*x^2 + 3*(88*a^2*b^6*c + 5*a^3*b^5*d)/b^6)*sqrt(b*x^2 + a)*x - 5/1
28*(8*a^3*b*c - a^4*d)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2)