3.81 \(\int \frac{\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=257 \[ -\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}+\frac{d \left (-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (-105 a^3 d^3+290 a^2 b c d^2-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt{a+b x^2}} \]

[Out]

-(d*(48*b^3*c^3 - 248*a*b^2*c^2*d + 290*a^2*b*c*d^2 - 105*a^3*d^3)*x*Sqrt[a + b*
x^2])/(48*a*b^4) - (d*(24*b^2*c^2 - 64*a*b*c*d + 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(
c + d*x^2))/(24*a*b^3) - (d*(6*b*c - 7*a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2)^2)/(6*
a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^3)/(a*b*Sqrt[a + b*x^2]) + (d*(64*b^3*c^3 -
144*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(16*b^(9/2))

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Rubi [A]  time = 0.632956, antiderivative size = 257, 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 \[ -\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}+\frac{d \left (-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (-105 a^3 d^3+290 a^2 b c d^2-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^4/(a + b*x^2)^(3/2),x]

[Out]

-(d*(48*b^3*c^3 - 248*a*b^2*c^2*d + 290*a^2*b*c*d^2 - 105*a^3*d^3)*x*Sqrt[a + b*
x^2])/(48*a*b^4) - (d*(24*b^2*c^2 - 64*a*b*c*d + 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(
c + d*x^2))/(24*a*b^3) - (d*(6*b*c - 7*a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2)^2)/(6*
a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^3)/(a*b*Sqrt[a + b*x^2]) + (d*(64*b^3*c^3 -
144*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(16*b^(9/2))

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Rubi in Sympy [A]  time = 85.0963, size = 262, normalized size = 1.02 \[ - \frac{d \left (35 a^{3} d^{3} - 120 a^{2} b c d^{2} + 144 a b^{2} c^{2} d - 64 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{9}{2}}} - \frac{x \left (c + d x^{2}\right )^{3} \left (a d - b c\right )}{a b \sqrt{a + b x^{2}}} + \frac{d x \sqrt{a + b x^{2}} \left (c + d x^{2}\right )^{2} \left (7 a d - 6 b c\right )}{6 a b^{2}} - \frac{d^{2} x \sqrt{a + b x^{2}} \left (a c \left (7 a d - 12 b c\right ) + x^{2} \left (35 a^{2} d^{2} - 64 a b c d + 24 b^{2} c^{2}\right )\right )}{24 a b^{3}} + \frac{d x \sqrt{a + b x^{2}} \left (105 a^{3} d^{3} - 346 a^{2} b c d^{2} + 352 a b^{2} c^{2} d - 96 b^{3} c^{3}\right )}{48 a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**4/(b*x**2+a)**(3/2),x)

[Out]

-d*(35*a**3*d**3 - 120*a**2*b*c*d**2 + 144*a*b**2*c**2*d - 64*b**3*c**3)*atanh(s
qrt(b)*x/sqrt(a + b*x**2))/(16*b**(9/2)) - x*(c + d*x**2)**3*(a*d - b*c)/(a*b*sq
rt(a + b*x**2)) + d*x*sqrt(a + b*x**2)*(c + d*x**2)**2*(7*a*d - 6*b*c)/(6*a*b**2
) - d**2*x*sqrt(a + b*x**2)*(a*c*(7*a*d - 12*b*c) + x**2*(35*a**2*d**2 - 64*a*b*
c*d + 24*b**2*c**2))/(24*a*b**3) + d*x*sqrt(a + b*x**2)*(105*a**3*d**3 - 346*a**
2*b*c*d**2 + 352*a*b**2*c**2*d - 96*b**3*c**3)/(48*a*b**4)

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Mathematica [A]  time = 0.400355, size = 172, normalized size = 0.67 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (3 d^2 \left (19 a^2 d^2-56 a b c d+48 b^2 c^2\right )+2 b d^3 x^2 (24 b c-11 a d)+\frac{48 (b c-a d)^4}{a \left (a+b x^2\right )}+8 b^2 d^4 x^4\right )+3 d \left (-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{48 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^4/(a + b*x^2)^(3/2),x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(3*d^2*(48*b^2*c^2 - 56*a*b*c*d + 19*a^2*d^2) + 2*b*d
^3*(24*b*c - 11*a*d)*x^2 + 8*b^2*d^4*x^4 + (48*(b*c - a*d)^4)/(a*(a + b*x^2))) +
3*d*(64*b^3*c^3 - 144*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*Log[b*x + Sqr
t[b]*Sqrt[a + b*x^2]])/(48*b^(9/2))

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Maple [A]  time = 0.024, size = 340, normalized size = 1.3 \[{\frac{{c}^{4}x}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{{d}^{4}{x}^{7}}{6\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{7\,a{d}^{4}{x}^{5}}{24\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{d}^{4}{a}^{2}{x}^{3}}{48\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{a}^{3}{d}^{4}x}{16\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,{a}^{3}{d}^{4}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{c{d}^{3}{x}^{5}}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,c{d}^{3}a{x}^{3}}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,c{d}^{3}{a}^{2}x}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,c{d}^{3}{a}^{2}}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+3\,{\frac{{c}^{2}{d}^{2}{x}^{3}}{b\sqrt{b{x}^{2}+a}}}+9\,{\frac{{c}^{2}{d}^{2}ax}{{b}^{2}\sqrt{b{x}^{2}+a}}}-9\,{\frac{{c}^{2}{d}^{2}a\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{5/2}}}-4\,{\frac{{c}^{3}dx}{b\sqrt{b{x}^{2}+a}}}+4\,{\frac{{c}^{3}d\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^4/(b*x^2+a)^(3/2),x)

[Out]

c^4*x/a/(b*x^2+a)^(1/2)+1/6*d^4*x^7/b/(b*x^2+a)^(1/2)-7/24*d^4*a/b^2*x^5/(b*x^2+
a)^(1/2)+35/48*d^4*a^2/b^3*x^3/(b*x^2+a)^(1/2)+35/16*d^4*a^3/b^4*x/(b*x^2+a)^(1/
2)-35/16*d^4*a^3/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+c*d^3*x^5/b/(b*x^2+a)^(1/
2)-5/2*c*d^3*a/b^2*x^3/(b*x^2+a)^(1/2)-15/2*c*d^3*a^2/b^3*x/(b*x^2+a)^(1/2)+15/2
*c*d^3*a^2/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+3*c^2*d^2*x^3/b/(b*x^2+a)^(1/2)
+9*c^2*d^2*a/b^2*x/(b*x^2+a)^(1/2)-9*c^2*d^2*a/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1
/2))-4*c^3*d*x/b/(b*x^2+a)^(1/2)+4*c^3*d/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((d*x^2 + c)^4/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.451806, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (8 \, a b^{3} d^{4} x^{7} + 2 \,{\left (24 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{5} +{\left (144 \, a b^{3} c^{2} d^{2} - 120 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{4} c^{4} - 64 \, a b^{3} c^{3} d + 144 \, a^{2} b^{2} c^{2} d^{2} - 120 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} +{\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \,{\left (a b^{5} x^{2} + a^{2} b^{4}\right )} \sqrt{b}}, \frac{{\left (8 \, a b^{3} d^{4} x^{7} + 2 \,{\left (24 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{5} +{\left (144 \, a b^{3} c^{2} d^{2} - 120 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{4} c^{4} - 64 \, a b^{3} c^{3} d + 144 \, a^{2} b^{2} c^{2} d^{2} - 120 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} +{\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \,{\left (a b^{5} x^{2} + a^{2} b^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^4/(b*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(2*(8*a*b^3*d^4*x^7 + 2*(24*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*x^5 + (144*a*b^3*
c^2*d^2 - 120*a^2*b^2*c*d^3 + 35*a^3*b*d^4)*x^3 + 3*(16*b^4*c^4 - 64*a*b^3*c^3*d
+ 144*a^2*b^2*c^2*d^2 - 120*a^3*b*c*d^3 + 35*a^4*d^4)*x)*sqrt(b*x^2 + a)*sqrt(b
) - 3*(64*a^2*b^3*c^3*d - 144*a^3*b^2*c^2*d^2 + 120*a^4*b*c*d^3 - 35*a^5*d^4 + (
64*a*b^4*c^3*d - 144*a^2*b^3*c^2*d^2 + 120*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^2)*lo
g(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((a*b^5*x^2 + a^2*b^4)*sqrt(b)
), 1/48*((8*a*b^3*d^4*x^7 + 2*(24*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*x^5 + (144*a*b^3*
c^2*d^2 - 120*a^2*b^2*c*d^3 + 35*a^3*b*d^4)*x^3 + 3*(16*b^4*c^4 - 64*a*b^3*c^3*d
+ 144*a^2*b^2*c^2*d^2 - 120*a^3*b*c*d^3 + 35*a^4*d^4)*x)*sqrt(b*x^2 + a)*sqrt(-
b) + 3*(64*a^2*b^3*c^3*d - 144*a^3*b^2*c^2*d^2 + 120*a^4*b*c*d^3 - 35*a^5*d^4 +
(64*a*b^4*c^3*d - 144*a^2*b^3*c^2*d^2 + 120*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^2)*a
rctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((a*b^5*x^2 + a^2*b^4)*sqrt(-b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**4/(b*x**2+a)**(3/2),x)

[Out]

Integral((c + d*x**2)**4/(a + b*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.248377, size = 317, normalized size = 1.23 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \, d^{4} x^{2}}{b} + \frac{24 \, a b^{6} c d^{3} - 7 \, a^{2} b^{5} d^{4}}{a b^{7}}\right )} x^{2} + \frac{144 \, a b^{6} c^{2} d^{2} - 120 \, a^{2} b^{5} c d^{3} + 35 \, a^{3} b^{4} d^{4}}{a b^{7}}\right )} x^{2} + \frac{3 \,{\left (16 \, b^{7} c^{4} - 64 \, a b^{6} c^{3} d + 144 \, a^{2} b^{5} c^{2} d^{2} - 120 \, a^{3} b^{4} c d^{3} + 35 \, a^{4} b^{3} d^{4}\right )}}{a b^{7}}\right )} x}{48 \, \sqrt{b x^{2} + a}} - \frac{{\left (64 \, b^{3} c^{3} d - 144 \, a b^{2} c^{2} d^{2} + 120 \, a^{2} b c d^{3} - 35 \, a^{3} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^4/(b*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/48*((2*(4*d^4*x^2/b + (24*a*b^6*c*d^3 - 7*a^2*b^5*d^4)/(a*b^7))*x^2 + (144*a*b
^6*c^2*d^2 - 120*a^2*b^5*c*d^3 + 35*a^3*b^4*d^4)/(a*b^7))*x^2 + 3*(16*b^7*c^4 -
64*a*b^6*c^3*d + 144*a^2*b^5*c^2*d^2 - 120*a^3*b^4*c*d^3 + 35*a^4*b^3*d^4)/(a*b^
7))*x/sqrt(b*x^2 + a) - 1/16*(64*b^3*c^3*d - 144*a*b^2*c^2*d^2 + 120*a^2*b*c*d^3
- 35*a^3*d^4)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)