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

Optimal. Leaf size=241 \[ \frac{x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac{a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac{a^2 x \sqrt{a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac{a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]

[Out]

(a^2*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(256*b^2) + (a*(80
*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(3/2))/(384*b^2) + ((80*b^2*c^2
- 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(5/2))/(480*b^2) + (3*d*(4*b*c - a*d)*x
*(a + b*x^2)^(7/2))/(80*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(10*b) + (a^3
*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(25
6*b^(5/2))

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Rubi [A]  time = 0.323662, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac{a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac{a^2 x \sqrt{a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac{a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(256*b^2) + (a*(80
*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(3/2))/(384*b^2) + ((80*b^2*c^2
- 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(5/2))/(480*b^2) + (3*d*(4*b*c - a*d)*x
*(a + b*x^2)^(7/2))/(80*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(10*b) + (a^3
*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(25
6*b^(5/2))

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Rubi in Sympy [A]  time = 32.758, size = 238, normalized size = 0.99 \[ \frac{a^{3} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{256 b^{\frac{5}{2}}} + \frac{a^{2} x \sqrt{a + b x^{2}} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right )}{256 b^{2}} + \frac{a x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right )}{384 b^{2}} + \frac{d x \left (a + b x^{2}\right )^{\frac{7}{2}} \left (c + d x^{2}\right )}{10 b} - \frac{3 d x \left (a + b x^{2}\right )^{\frac{7}{2}} \left (a d - 4 b c\right )}{80 b^{2}} + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (3 a^{2} d^{2} - 20 a b c d + 80 b^{2} c^{2}\right )}{480 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*(3*a**2*d**2 - 20*a*b*c*d + 80*b**2*c**2)*atanh(sqrt(b)*x/sqrt(a + b*x**2))
/(256*b**(5/2)) + a**2*x*sqrt(a + b*x**2)*(3*a**2*d**2 - 20*a*b*c*d + 80*b**2*c*
*2)/(256*b**2) + a*x*(a + b*x**2)**(3/2)*(3*a**2*d**2 - 20*a*b*c*d + 80*b**2*c**
2)/(384*b**2) + d*x*(a + b*x**2)**(7/2)*(c + d*x**2)/(10*b) - 3*d*x*(a + b*x**2)
**(7/2)*(a*d - 4*b*c)/(80*b**2) + x*(a + b*x**2)**(5/2)*(3*a**2*d**2 - 20*a*b*c*
d + 80*b**2*c**2)/(480*b**2)

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Mathematica [A]  time = 0.211399, size = 191, normalized size = 0.79 \[ \frac{15 a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} x \sqrt{a+b x^2} \left (-45 a^4 d^2+30 a^3 b d \left (10 c+d x^2\right )+8 a^2 b^2 \left (330 c^2+295 c d x^2+93 d^2 x^4\right )+16 a b^3 x^2 \left (130 c^2+170 c d x^2+63 d^2 x^4\right )+64 b^4 x^4 \left (10 c^2+15 c d x^2+6 d^2 x^4\right )\right )}{3840 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-45*a^4*d^2 + 30*a^3*b*d*(10*c + d*x^2) + 64*b^4*x^4
*(10*c^2 + 15*c*d*x^2 + 6*d^2*x^4) + 16*a*b^3*x^2*(130*c^2 + 170*c*d*x^2 + 63*d^
2*x^4) + 8*a^2*b^2*(330*c^2 + 295*c*d*x^2 + 93*d^2*x^4)) + 15*a^3*(80*b^2*c^2 -
20*a*b*c*d + 3*a^2*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(3840*b^(5/2))

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Maple [A]  time = 0.011, size = 308, normalized size = 1.3 \[{\frac{{c}^{2}x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a{c}^{2}x}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{2}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{c}^{2}{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{d}^{2}{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,a{d}^{2}x}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}{d}^{2}x}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{d}^{2}{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cdx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{acdx}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,cd{a}^{2}x}{96\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,cd{a}^{3}x}{64\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,cd{a}^{4}}{64}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*c^2*x*(b*x^2+a)^(5/2)+5/24*c^2*a*x*(b*x^2+a)^(3/2)+5/16*c^2*a^2*x*(b*x^2+a)^
(1/2)+5/16*c^2*a^3/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/10*d^2*x^3*(b*x^2+a)^
(7/2)/b-3/80*d^2*a/b^2*x*(b*x^2+a)^(7/2)+1/160*d^2*a^2/b^2*x*(b*x^2+a)^(5/2)+1/1
28*d^2*a^3/b^2*x*(b*x^2+a)^(3/2)+3/256*d^2*a^4/b^2*x*(b*x^2+a)^(1/2)+3/256*d^2*a
^5/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/4*c*d*x*(b*x^2+a)^(7/2)/b-1/24*c*d*a/
b*x*(b*x^2+a)^(5/2)-5/96*c*d*a^2/b*x*(b*x^2+a)^(3/2)-5/64*c*d*a^3/b*x*(b*x^2+a)^
(1/2)-5/64*c*d*a^4/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.556019, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, b^{4} d^{2} x^{9} + 48 \,{\left (20 \, b^{4} c d + 21 \, a b^{3} d^{2}\right )} x^{7} + 8 \,{\left (80 \, b^{4} c^{2} + 340 \, a b^{3} c d + 93 \, a^{2} b^{2} d^{2}\right )} x^{5} + 10 \,{\left (208 \, a b^{3} c^{2} + 236 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + 15 \,{\left (176 \, a^{2} b^{2} c^{2} + 20 \, a^{3} b c d - 3 \, a^{4} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} + 15 \,{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{7680 \, b^{\frac{5}{2}}}, \frac{{\left (384 \, b^{4} d^{2} x^{9} + 48 \,{\left (20 \, b^{4} c d + 21 \, a b^{3} d^{2}\right )} x^{7} + 8 \,{\left (80 \, b^{4} c^{2} + 340 \, a b^{3} c d + 93 \, a^{2} b^{2} d^{2}\right )} x^{5} + 10 \,{\left (208 \, a b^{3} c^{2} + 236 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + 15 \,{\left (176 \, a^{2} b^{2} c^{2} + 20 \, a^{3} b c d - 3 \, a^{4} d^{2}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{3840 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(2*(384*b^4*d^2*x^9 + 48*(20*b^4*c*d + 21*a*b^3*d^2)*x^7 + 8*(80*b^4*c^2
+ 340*a*b^3*c*d + 93*a^2*b^2*d^2)*x^5 + 10*(208*a*b^3*c^2 + 236*a^2*b^2*c*d + 3
*a^3*b*d^2)*x^3 + 15*(176*a^2*b^2*c^2 + 20*a^3*b*c*d - 3*a^4*d^2)*x)*sqrt(b*x^2
+ a)*sqrt(b) + 15*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*log(-2*sqrt(b*x^2
+ a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(5/2), 1/3840*((384*b^4*d^2*x^9 + 48*(20*b^
4*c*d + 21*a*b^3*d^2)*x^7 + 8*(80*b^4*c^2 + 340*a*b^3*c*d + 93*a^2*b^2*d^2)*x^5
+ 10*(208*a*b^3*c^2 + 236*a^2*b^2*c*d + 3*a^3*b*d^2)*x^3 + 15*(176*a^2*b^2*c^2 +
20*a^3*b*c*d - 3*a^4*d^2)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 15*(80*a^3*b^2*c^2 - 20
*a^4*b*c*d + 3*a^5*d^2)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 170.682, size = 537, normalized size = 2.23 \[ - \frac{3 a^{\frac{9}{2}} d^{2} x}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{7}{2}} c d x}{64 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{7}{2}} d^{2} x^{3}}{256 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{5}{2}} c^{2} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 a^{\frac{5}{2}} c^{2} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 a^{\frac{5}{2}} c d x^{3}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{129 a^{\frac{5}{2}} d^{2} x^{5}}{640 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{\frac{3}{2}} b c^{2} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 a^{\frac{3}{2}} b c d x^{5}}{96 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{73 a^{\frac{3}{2}} b d^{2} x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 \sqrt{a} b^{2} c^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{2} c d x^{7}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{29 \sqrt{a} b^{2} d^{2} x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{5} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{5}{2}}} - \frac{5 a^{4} c d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{5 a^{3} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{b^{3} c^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} c d x^{9}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b^{3} d^{2} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**(9/2)*d**2*x/(256*b**2*sqrt(1 + b*x**2/a)) + 5*a**(7/2)*c*d*x/(64*b*sqrt(1
+ b*x**2/a)) - a**(7/2)*d**2*x**3/(256*b*sqrt(1 + b*x**2/a)) + a**(5/2)*c**2*x*
sqrt(1 + b*x**2/a)/2 + 3*a**(5/2)*c**2*x/(16*sqrt(1 + b*x**2/a)) + 133*a**(5/2)*
c*d*x**3/(192*sqrt(1 + b*x**2/a)) + 129*a**(5/2)*d**2*x**5/(640*sqrt(1 + b*x**2/
a)) + 35*a**(3/2)*b*c**2*x**3/(48*sqrt(1 + b*x**2/a)) + 127*a**(3/2)*b*c*d*x**5/
(96*sqrt(1 + b*x**2/a)) + 73*a**(3/2)*b*d**2*x**7/(160*sqrt(1 + b*x**2/a)) + 17*
sqrt(a)*b**2*c**2*x**5/(24*sqrt(1 + b*x**2/a)) + 23*sqrt(a)*b**2*c*d*x**7/(24*sq
rt(1 + b*x**2/a)) + 29*sqrt(a)*b**2*d**2*x**9/(80*sqrt(1 + b*x**2/a)) + 3*a**5*d
**2*asinh(sqrt(b)*x/sqrt(a))/(256*b**(5/2)) - 5*a**4*c*d*asinh(sqrt(b)*x/sqrt(a)
)/(64*b**(3/2)) + 5*a**3*c**2*asinh(sqrt(b)*x/sqrt(a))/(16*sqrt(b)) + b**3*c**2*
x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + b**3*c*d*x**9/(4*sqrt(a)*sqrt(1 + b*x**2/a
)) + b**3*d**2*x**11/(10*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.366916, size = 298, normalized size = 1.24 \[ \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b^{2} d^{2} x^{2} + \frac{20 \, b^{10} c d + 21 \, a b^{9} d^{2}}{b^{8}}\right )} x^{2} + \frac{80 \, b^{10} c^{2} + 340 \, a b^{9} c d + 93 \, a^{2} b^{8} d^{2}}{b^{8}}\right )} x^{2} + \frac{5 \,{\left (208 \, a b^{9} c^{2} + 236 \, a^{2} b^{8} c d + 3 \, a^{3} b^{7} d^{2}\right )}}{b^{8}}\right )} x^{2} + \frac{15 \,{\left (176 \, a^{2} b^{8} c^{2} + 20 \, a^{3} b^{7} c d - 3 \, a^{4} b^{6} d^{2}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*(2*(4*(6*(8*b^2*d^2*x^2 + (20*b^10*c*d + 21*a*b^9*d^2)/b^8)*x^2 + (80*b^1
0*c^2 + 340*a*b^9*c*d + 93*a^2*b^8*d^2)/b^8)*x^2 + 5*(208*a*b^9*c^2 + 236*a^2*b^
8*c*d + 3*a^3*b^7*d^2)/b^8)*x^2 + 15*(176*a^2*b^8*c^2 + 20*a^3*b^7*c*d - 3*a^4*b
^6*d^2)/b^8)*sqrt(b*x^2 + a)*x - 1/256*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^
2)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)