3.89 \(\int \frac{\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{x \left (c+d x^2\right )^2 (b c-a d) (7 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}+\frac{d^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{12 a^2 b^3}-\frac{d x \sqrt{a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 b^4}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]

[Out]

-(d*(16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x
^2])/(24*a^2*b^4) - (d*(8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(
c + d*x^2))/(12*a^2*b^3) + ((b*c - a*d)*(2*b*c + 7*a*d)*x*(c + d*x^2)^2)/(3*a^2*
b^2*Sqrt[a + b*x^2]) + ((b*c - a*d)*x*(c + d*x^2)^3)/(3*a*b*(a + b*x^2)^(3/2)) +
(d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]
])/(8*b^(9/2))

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Rubi [A]  time = 0.614599, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x \left (c+d x^2\right )^2 (b c-a d) (7 a d+2 b c)}{3 a^2 b^2 \sqrt{a+b x^2}}+\frac{d^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{12 a^2 b^3}-\frac{d x \sqrt{a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 b^4}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^4/(a + b*x^2)^(5/2),x]

[Out]

-(d*(16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x
^2])/(24*a^2*b^4) - (d*(8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(
c + d*x^2))/(12*a^2*b^3) + ((b*c - a*d)*(2*b*c + 7*a*d)*x*(c + d*x^2)^2)/(3*a^2*
b^2*Sqrt[a + b*x^2]) + ((b*c - a*d)*x*(c + d*x^2)^3)/(3*a*b*(a + b*x^2)^(3/2)) +
(d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]
])/(8*b^(9/2))

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Rubi in Sympy [A]  time = 91.4008, size = 260, normalized size = 1.02 \[ \frac{d^{2} \left (35 a^{2} d^{2} - 80 a b c d + 48 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{9}{2}}} - \frac{x \left (c + d x^{2}\right )^{3} \left (a d - b c\right )}{3 a b \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{x \left (c + d x^{2}\right )^{2} \left (a d - b c\right ) \left (7 a d + 2 b c\right )}{3 a^{2} b^{2} \sqrt{a + b x^{2}}} + \frac{d^{2} x \sqrt{a + b x^{2}} \left (a c \left (7 a d - 4 b c\right ) + x^{2} \left (35 a^{2} d^{2} - 24 a b c d - 8 b^{2} c^{2}\right )\right )}{12 a^{2} b^{3}} - \frac{d x \sqrt{a + b x^{2}} \left (105 a^{3} d^{3} - 226 a^{2} b c d^{2} + 80 a b^{2} c^{2} d + 32 b^{3} c^{3}\right )}{24 a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**4/(b*x**2+a)**(5/2),x)

[Out]

d**2*(35*a**2*d**2 - 80*a*b*c*d + 48*b**2*c**2)*atanh(sqrt(b)*x/sqrt(a + b*x**2)
)/(8*b**(9/2)) - x*(c + d*x**2)**3*(a*d - b*c)/(3*a*b*(a + b*x**2)**(3/2)) - x*(
c + d*x**2)**2*(a*d - b*c)*(7*a*d + 2*b*c)/(3*a**2*b**2*sqrt(a + b*x**2)) + d**2
*x*sqrt(a + b*x**2)*(a*c*(7*a*d - 4*b*c) + x**2*(35*a**2*d**2 - 24*a*b*c*d - 8*b
**2*c**2))/(12*a**2*b**3) - d*x*sqrt(a + b*x**2)*(105*a**3*d**3 - 226*a**2*b*c*d
**2 + 80*a*b**2*c**2*d + 32*b**3*c**3)/(24*a**2*b**4)

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Mathematica [A]  time = 0.277545, size = 157, normalized size = 0.62 \[ \frac{x \sqrt{a+b x^2} \left (\frac{16 (b c-a d)^3 (5 a d+b c)}{a^2 \left (a+b x^2\right )}+3 d^3 (16 b c-11 a d)+\frac{8 (b c-a d)^4}{a \left (a+b x^2\right )^2}+6 b d^4 x^2\right )}{24 b^4}+\frac{d^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^4/(a + b*x^2)^(5/2),x]

[Out]

(x*Sqrt[a + b*x^2]*(3*d^3*(16*b*c - 11*a*d) + 6*b*d^4*x^2 + (8*(b*c - a*d)^4)/(a
*(a + b*x^2)^2) + (16*(b*c - a*d)^3*(b*c + 5*a*d))/(a^2*(a + b*x^2))))/(24*b^4)
+ (d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]
)/(8*b^(9/2))

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Maple [A]  time = 0.024, size = 351, normalized size = 1.4 \[{\frac{{c}^{4}x}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,{c}^{4}x}{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{{d}^{4}{x}^{7}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,a{d}^{4}{x}^{5}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{a}^{2}{d}^{4}{x}^{3}}{24\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{a}^{2}{d}^{4}x}{8\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{a}^{2}{d}^{4}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+2\,{\frac{c{d}^{3}{x}^{5}}{b \left ( b{x}^{2}+a \right ) ^{3/2}}}+{\frac{10\,ac{d}^{3}{x}^{3}}{3\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+10\,{\frac{ac{d}^{3}x}{{b}^{3}\sqrt{b{x}^{2}+a}}}-10\,{\frac{ac{d}^{3}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{7/2}}}-2\,{\frac{{c}^{2}{d}^{2}{x}^{3}}{b \left ( b{x}^{2}+a \right ) ^{3/2}}}-6\,{\frac{{c}^{2}{d}^{2}x}{{b}^{2}\sqrt{b{x}^{2}+a}}}+6\,{\frac{{c}^{2}{d}^{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{5/2}}}-{\frac{4\,{c}^{3}dx}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,{c}^{3}dx}{3\,ab}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^4/(b*x^2+a)^(5/2),x)

[Out]

1/3*c^4*x/a/(b*x^2+a)^(3/2)+2/3*c^4/a^2*x/(b*x^2+a)^(1/2)+1/4*d^4*x^7/b/(b*x^2+a
)^(3/2)-7/8*d^4*a/b^2*x^5/(b*x^2+a)^(3/2)-35/24*d^4*a^2/b^3*x^3/(b*x^2+a)^(3/2)-
35/8*d^4*a^2/b^4*x/(b*x^2+a)^(1/2)+35/8*d^4*a^2/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(
1/2))+2*c*d^3*x^5/b/(b*x^2+a)^(3/2)+10/3*c*d^3*a/b^2*x^3/(b*x^2+a)^(3/2)+10*c*d^
3*a/b^3*x/(b*x^2+a)^(1/2)-10*c*d^3*a/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-2*c^2
*d^2*x^3/b/(b*x^2+a)^(3/2)-6*c^2*d^2/b^2*x/(b*x^2+a)^(1/2)+6*c^2*d^2/b^(5/2)*ln(
x*b^(1/2)+(b*x^2+a)^(1/2))-4/3*c^3*d/b*x/(b*x^2+a)^(3/2)+4/3*c^3*d/a/b*x/(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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.561206, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (6 \, a^{2} b^{3} d^{4} x^{7} + 3 \,{\left (16 \, a^{2} b^{3} c d^{3} - 7 \, a^{3} b^{2} d^{4}\right )} x^{5} + 4 \,{\left (4 \, b^{5} c^{4} + 8 \, a b^{4} c^{3} d - 48 \, a^{2} b^{3} c^{2} d^{2} + 80 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{3} + 3 \,{\left (8 \, a b^{4} c^{4} - 48 \, a^{3} b^{2} c^{2} d^{2} + 80 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} + 3 \,{\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} +{\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \,{\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{48 \,{\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \sqrt{b}}, \frac{{\left (6 \, a^{2} b^{3} d^{4} x^{7} + 3 \,{\left (16 \, a^{2} b^{3} c d^{3} - 7 \, a^{3} b^{2} d^{4}\right )} x^{5} + 4 \,{\left (4 \, b^{5} c^{4} + 8 \, a b^{4} c^{3} d - 48 \, a^{2} b^{3} c^{2} d^{2} + 80 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{3} + 3 \,{\left (8 \, a b^{4} c^{4} - 48 \, a^{3} b^{2} c^{2} d^{2} + 80 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} +{\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \,{\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{24 \,{\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^4/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(2*(6*a^2*b^3*d^4*x^7 + 3*(16*a^2*b^3*c*d^3 - 7*a^3*b^2*d^4)*x^5 + 4*(4*b^
5*c^4 + 8*a*b^4*c^3*d - 48*a^2*b^3*c^2*d^2 + 80*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^
3 + 3*(8*a*b^4*c^4 - 48*a^3*b^2*c^2*d^2 + 80*a^4*b*c*d^3 - 35*a^5*d^4)*x)*sqrt(b
*x^2 + a)*sqrt(b) + 3*(48*a^4*b^2*c^2*d^2 - 80*a^5*b*c*d^3 + 35*a^6*d^4 + (48*a^
2*b^4*c^2*d^2 - 80*a^3*b^3*c*d^3 + 35*a^4*b^2*d^4)*x^4 + 2*(48*a^3*b^3*c^2*d^2 -
80*a^4*b^2*c*d^3 + 35*a^5*b*d^4)*x^2)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a
)*sqrt(b)))/((a^2*b^6*x^4 + 2*a^3*b^5*x^2 + a^4*b^4)*sqrt(b)), 1/24*((6*a^2*b^3*
d^4*x^7 + 3*(16*a^2*b^3*c*d^3 - 7*a^3*b^2*d^4)*x^5 + 4*(4*b^5*c^4 + 8*a*b^4*c^3*
d - 48*a^2*b^3*c^2*d^2 + 80*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^3 + 3*(8*a*b^4*c^4 -
48*a^3*b^2*c^2*d^2 + 80*a^4*b*c*d^3 - 35*a^5*d^4)*x)*sqrt(b*x^2 + a)*sqrt(-b) +
3*(48*a^4*b^2*c^2*d^2 - 80*a^5*b*c*d^3 + 35*a^6*d^4 + (48*a^2*b^4*c^2*d^2 - 80*
a^3*b^3*c*d^3 + 35*a^4*b^2*d^4)*x^4 + 2*(48*a^3*b^3*c^2*d^2 - 80*a^4*b^2*c*d^3 +
35*a^5*b*d^4)*x^2)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((a^2*b^6*x^4 + 2*a^3*b^
5*x^2 + a^4*b^4)*sqrt(-b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**4/(b*x**2+a)**(5/2),x)

[Out]

Integral((c + d*x**2)**4/(a + b*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.234234, size = 320, normalized size = 1.25 \[ \frac{{\left ({\left (3 \,{\left (\frac{2 \, d^{4} x^{2}}{b} + \frac{16 \, a^{2} b^{6} c d^{3} - 7 \, a^{3} b^{5} d^{4}}{a^{2} b^{7}}\right )} x^{2} + \frac{4 \,{\left (4 \, b^{8} c^{4} + 8 \, a b^{7} c^{3} d - 48 \, a^{2} b^{6} c^{2} d^{2} + 80 \, a^{3} b^{5} c d^{3} - 35 \, a^{4} b^{4} d^{4}\right )}}{a^{2} b^{7}}\right )} x^{2} + \frac{3 \,{\left (8 \, a b^{7} c^{4} - 48 \, a^{3} b^{5} c^{2} d^{2} + 80 \, a^{4} b^{4} c d^{3} - 35 \, a^{5} b^{3} d^{4}\right )}}{a^{2} b^{7}}\right )} x}{24 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{{\left (48 \, b^{2} c^{2} d^{2} - 80 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^4/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

1/24*((3*(2*d^4*x^2/b + (16*a^2*b^6*c*d^3 - 7*a^3*b^5*d^4)/(a^2*b^7))*x^2 + 4*(4
*b^8*c^4 + 8*a*b^7*c^3*d - 48*a^2*b^6*c^2*d^2 + 80*a^3*b^5*c*d^3 - 35*a^4*b^4*d^
4)/(a^2*b^7))*x^2 + 3*(8*a*b^7*c^4 - 48*a^3*b^5*c^2*d^2 + 80*a^4*b^4*c*d^3 - 35*
a^5*b^3*d^4)/(a^2*b^7))*x/(b*x^2 + a)^(3/2) - 1/8*(48*b^2*c^2*d^2 - 80*a*b*c*d^3
+ 35*a^2*d^4)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)