3.617 \(\int \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=196 \[ \frac{x \left (c+d x^2\right )^{3/2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{192 d^2}+\frac{c x \sqrt{c+d x^2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{128 d^2}+\frac{c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{5/2}}-\frac{b x \left (c+d x^2\right )^{5/2} (3 b c-10 a d)}{48 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d} \]

[Out]

(c*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*x*Sqrt[c + d*x^2])/(128*d^2) + ((3*b^2*
c^2 - 16*a*b*c*d + 48*a^2*d^2)*x*(c + d*x^2)^(3/2))/(192*d^2) - (b*(3*b*c - 10*a
*d)*x*(c + d*x^2)^(5/2))/(48*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(5/2))/(8*d) +
(c^2*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])
/(128*d^(5/2))

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Rubi [A]  time = 0.270342, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (c+d x^2\right )^{3/2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{192 d^2}+\frac{c x \sqrt{c+d x^2} \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{128 d^2}+\frac{c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{5/2}}-\frac{b x \left (c+d x^2\right )^{5/2} (3 b c-10 a d)}{48 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{8 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2*(c + d*x^2)^(3/2),x]

[Out]

(c*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*x*Sqrt[c + d*x^2])/(128*d^2) + ((3*b^2*
c^2 - 16*a*b*c*d + 48*a^2*d^2)*x*(c + d*x^2)^(3/2))/(192*d^2) - (b*(3*b*c - 10*a
*d)*x*(c + d*x^2)^(5/2))/(48*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(5/2))/(8*d) +
(c^2*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])
/(128*d^(5/2))

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Rubi in Sympy [A]  time = 27.3776, size = 192, normalized size = 0.98 \[ \frac{b x \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}{8 d} + \frac{b x \left (c + d x^{2}\right )^{\frac{5}{2}} \left (10 a d - 3 b c\right )}{48 d^{2}} + \frac{c^{2} \left (48 a^{2} d^{2} - 16 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{128 d^{\frac{5}{2}}} + \frac{c x \sqrt{c + d x^{2}} \left (48 a^{2} d^{2} - 16 a b c d + 3 b^{2} c^{2}\right )}{128 d^{2}} + \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (48 a^{2} d^{2} - 16 a b c d + 3 b^{2} c^{2}\right )}{192 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(3/2),x)

[Out]

b*x*(a + b*x**2)*(c + d*x**2)**(5/2)/(8*d) + b*x*(c + d*x**2)**(5/2)*(10*a*d - 3
*b*c)/(48*d**2) + c**2*(48*a**2*d**2 - 16*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(d)*x
/sqrt(c + d*x**2))/(128*d**(5/2)) + c*x*sqrt(c + d*x**2)*(48*a**2*d**2 - 16*a*b*
c*d + 3*b**2*c**2)/(128*d**2) + x*(c + d*x**2)**(3/2)*(48*a**2*d**2 - 16*a*b*c*d
+ 3*b**2*c**2)/(192*d**2)

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Mathematica [A]  time = 0.163647, size = 159, normalized size = 0.81 \[ \frac{3 c^2 \left (48 a^2 d^2-16 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (48 a^2 d^2 \left (5 c+2 d x^2\right )+16 a b d \left (3 c^2+14 c d x^2+8 d^2 x^4\right )+b^2 \left (-9 c^3+6 c^2 d x^2+72 c d^2 x^4+48 d^3 x^6\right )\right )}{384 d^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2*(c + d*x^2)^(3/2),x]

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(48*a^2*d^2*(5*c + 2*d*x^2) + 16*a*b*d*(3*c^2 + 14*c*
d*x^2 + 8*d^2*x^4) + b^2*(-9*c^3 + 6*c^2*d*x^2 + 72*c*d^2*x^4 + 48*d^3*x^6)) + 3
*c^2*(3*b^2*c^2 - 16*a*b*c*d + 48*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(
384*d^(5/2))

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Maple [A]  time = 0.011, size = 249, normalized size = 1.3 \[{\frac{{a}^{2}x}{4} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}cx}{8}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{{b}^{2}{x}^{3}}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}cx}{16\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{64\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,x{b}^{2}{c}^{3}}{128\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{abx}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abcx}{12\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{ab{c}^{2}x}{8\,d}\sqrt{d{x}^{2}+c}}-{\frac{ab{c}^{3}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(3/2),x)

[Out]

1/4*a^2*x*(d*x^2+c)^(3/2)+3/8*a^2*c*x*(d*x^2+c)^(1/2)+3/8*a^2*c^2/d^(1/2)*ln(x*d
^(1/2)+(d*x^2+c)^(1/2))+1/8*b^2*x^3*(d*x^2+c)^(5/2)/d-1/16*b^2*c/d^2*x*(d*x^2+c)
^(5/2)+1/64*b^2*c^2/d^2*x*(d*x^2+c)^(3/2)+3/128*b^2*c^3/d^2*x*(d*x^2+c)^(1/2)+3/
128*b^2*c^4/d^(5/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/3*a*b*x*(d*x^2+c)^(5/2)/d-1/
12*a*b*c/d*x*(d*x^2+c)^(3/2)-1/8*a*b*c^2/d*x*(d*x^2+c)^(1/2)-1/8*a*b*c^3/d^(3/2)
*ln(x*d^(1/2)+(d*x^2+c)^(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)^2*(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.327499, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, b^{2} d^{3} x^{7} + 8 \,{\left (9 \, b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{5} + 2 \,{\left (3 \, b^{2} c^{2} d + 112 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (3 \, b^{2} c^{3} - 16 \, a b c^{2} d - 80 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{768 \, d^{\frac{5}{2}}}, \frac{{\left (48 \, b^{2} d^{3} x^{7} + 8 \,{\left (9 \, b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{5} + 2 \,{\left (3 \, b^{2} c^{2} d + 112 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (3 \, b^{2} c^{3} - 16 \, a b c^{2} d - 80 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 3 \,{\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{384 \, \sqrt{-d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2),x, algorithm="fricas")

[Out]

[1/768*(2*(48*b^2*d^3*x^7 + 8*(9*b^2*c*d^2 + 16*a*b*d^3)*x^5 + 2*(3*b^2*c^2*d +
112*a*b*c*d^2 + 48*a^2*d^3)*x^3 - 3*(3*b^2*c^3 - 16*a*b*c^2*d - 80*a^2*c*d^2)*x)
*sqrt(d*x^2 + c)*sqrt(d) + 3*(3*b^2*c^4 - 16*a*b*c^3*d + 48*a^2*c^2*d^2)*log(-2*
sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/d^(5/2), 1/384*((48*b^2*d^3*x^7 +
8*(9*b^2*c*d^2 + 16*a*b*d^3)*x^5 + 2*(3*b^2*c^2*d + 112*a*b*c*d^2 + 48*a^2*d^3)*
x^3 - 3*(3*b^2*c^3 - 16*a*b*c^2*d - 80*a^2*c*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d) +
3*(3*b^2*c^4 - 16*a*b*c^3*d + 48*a^2*c^2*d^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c))
)/(sqrt(-d)*d^2)]

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Sympy [A]  time = 88.1589, size = 440, normalized size = 2.24 \[ \frac{a^{2} c^{\frac{3}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{a^{2} c^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} \sqrt{c} d x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{d}} + \frac{a^{2} d^{2} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{a b c^{\frac{5}{2}} x}{8 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a b c^{\frac{3}{2}} x^{3}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{11 a b \sqrt{c} d x^{5}}{12 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{3}{2}}} + \frac{a b d^{2} x^{7}}{3 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 b^{2} c^{\frac{7}{2}} x}{128 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{5}{2}} x^{3}}{128 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{13 b^{2} c^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} \sqrt{c} d x^{7}}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{5}{2}}} + \frac{b^{2} d^{2} x^{9}}{8 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(3/2),x)

[Out]

a**2*c**(3/2)*x*sqrt(1 + d*x**2/c)/2 + a**2*c**(3/2)*x/(8*sqrt(1 + d*x**2/c)) +
3*a**2*sqrt(c)*d*x**3/(8*sqrt(1 + d*x**2/c)) + 3*a**2*c**2*asinh(sqrt(d)*x/sqrt(
c))/(8*sqrt(d)) + a**2*d**2*x**5/(4*sqrt(c)*sqrt(1 + d*x**2/c)) + a*b*c**(5/2)*x
/(8*d*sqrt(1 + d*x**2/c)) + 17*a*b*c**(3/2)*x**3/(24*sqrt(1 + d*x**2/c)) + 11*a*
b*sqrt(c)*d*x**5/(12*sqrt(1 + d*x**2/c)) - a*b*c**3*asinh(sqrt(d)*x/sqrt(c))/(8*
d**(3/2)) + a*b*d**2*x**7/(3*sqrt(c)*sqrt(1 + d*x**2/c)) - 3*b**2*c**(7/2)*x/(12
8*d**2*sqrt(1 + d*x**2/c)) - b**2*c**(5/2)*x**3/(128*d*sqrt(1 + d*x**2/c)) + 13*
b**2*c**(3/2)*x**5/(64*sqrt(1 + d*x**2/c)) + 5*b**2*sqrt(c)*d*x**7/(16*sqrt(1 +
d*x**2/c)) + 3*b**2*c**4*asinh(sqrt(d)*x/sqrt(c))/(128*d**(5/2)) + b**2*d**2*x**
9/(8*sqrt(c)*sqrt(1 + d*x**2/c))

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GIAC/XCAS [A]  time = 0.235794, size = 236, normalized size = 1.2 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, b^{2} d x^{2} + \frac{9 \, b^{2} c d^{6} + 16 \, a b d^{7}}{d^{6}}\right )} x^{2} + \frac{3 \, b^{2} c^{2} d^{5} + 112 \, a b c d^{6} + 48 \, a^{2} d^{7}}{d^{6}}\right )} x^{2} - \frac{3 \,{\left (3 \, b^{2} c^{3} d^{4} - 16 \, a b c^{2} d^{5} - 80 \, a^{2} c d^{6}\right )}}{d^{6}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (3 \, b^{2} c^{4} - 16 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{128 \, d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*b^2*d*x^2 + (9*b^2*c*d^6 + 16*a*b*d^7)/d^6)*x^2 + (3*b^2*c^2*d^5
+ 112*a*b*c*d^6 + 48*a^2*d^7)/d^6)*x^2 - 3*(3*b^2*c^3*d^4 - 16*a*b*c^2*d^5 - 80*
a^2*c*d^6)/d^6)*sqrt(d*x^2 + c)*x - 1/128*(3*b^2*c^4 - 16*a*b*c^3*d + 48*a^2*c^2
*d^2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(5/2)