3.151 \(\int \frac{x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=245 \[ \frac{x^5 \sqrt{a+b x^2} \left (63 a^2 f-70 a b e+80 b^2 d\right )}{480 b^3}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^{11/2}}-\frac{a x \sqrt{a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^5}+\frac{x^3 \sqrt{a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{384 b^4}+\frac{x^7 \sqrt{a+b x^2} (10 b e-9 a f)}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b} \]

[Out]

-(a*(96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*x*Sqrt[a + b*x^2])/(256*b^5)
+ ((96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*x^3*Sqrt[a + b*x^2])/(384*b^
4) + ((80*b^2*d - 70*a*b*e + 63*a^2*f)*x^5*Sqrt[a + b*x^2])/(480*b^3) + ((10*b*e
- 9*a*f)*x^7*Sqrt[a + b*x^2])/(80*b^2) + (f*x^9*Sqrt[a + b*x^2])/(10*b) + (a^2*
(96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(256*b^(11/2))

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Rubi [A]  time = 0.649824, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{x^5 \sqrt{a+b x^2} \left (63 a^2 f-70 a b e+80 b^2 d\right )}{480 b^3}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^{11/2}}-\frac{a x \sqrt{a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^5}+\frac{x^3 \sqrt{a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{384 b^4}+\frac{x^7 \sqrt{a+b x^2} (10 b e-9 a f)}{80 b^2}+\frac{f x^9 \sqrt{a+b x^2}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

-(a*(96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*x*Sqrt[a + b*x^2])/(256*b^5)
+ ((96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*x^3*Sqrt[a + b*x^2])/(384*b^
4) + ((80*b^2*d - 70*a*b*e + 63*a^2*f)*x^5*Sqrt[a + b*x^2])/(480*b^3) + ((10*b*e
- 9*a*f)*x^7*Sqrt[a + b*x^2])/(80*b^2) + (f*x^9*Sqrt[a + b*x^2])/(10*b) + (a^2*
(96*b^3*c - 80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(256*b^(11/2))

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Rubi in Sympy [A]  time = 70.8295, size = 238, normalized size = 0.97 \[ - \frac{a^{2} \left (a \left (63 a^{2} f - 70 a b e + 80 b^{2} d\right ) - 96 b^{3} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{256 b^{\frac{11}{2}}} + \frac{a x \sqrt{a + b x^{2}} \left (a \left (63 a^{2} f - 70 a b e + 80 b^{2} d\right ) - 96 b^{3} c\right )}{256 b^{5}} + \frac{f x^{9} \sqrt{a + b x^{2}}}{10 b} - \frac{x^{7} \sqrt{a + b x^{2}} \left (9 a f - 10 b e\right )}{80 b^{2}} + \frac{x^{5} \sqrt{a + b x^{2}} \left (63 a^{2} f - 70 a b e + 80 b^{2} d\right )}{480 b^{3}} - \frac{x^{3} \sqrt{a + b x^{2}} \left (a \left (63 a^{2} f - 70 a b e + 80 b^{2} d\right ) - 96 b^{3} c\right )}{384 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

-a**2*(a*(63*a**2*f - 70*a*b*e + 80*b**2*d) - 96*b**3*c)*atanh(sqrt(b)*x/sqrt(a
+ b*x**2))/(256*b**(11/2)) + a*x*sqrt(a + b*x**2)*(a*(63*a**2*f - 70*a*b*e + 80*
b**2*d) - 96*b**3*c)/(256*b**5) + f*x**9*sqrt(a + b*x**2)/(10*b) - x**7*sqrt(a +
b*x**2)*(9*a*f - 10*b*e)/(80*b**2) + x**5*sqrt(a + b*x**2)*(63*a**2*f - 70*a*b*
e + 80*b**2*d)/(480*b**3) - x**3*sqrt(a + b*x**2)*(a*(63*a**2*f - 70*a*b*e + 80*
b**2*d) - 96*b**3*c)/(384*b**4)

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Mathematica [A]  time = 0.242686, size = 188, normalized size = 0.77 \[ \frac{15 a^2 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )-\sqrt{b} x \sqrt{a+b x^2} \left (-945 a^4 f+210 a^3 b \left (5 e+3 f x^2\right )-4 a^2 b^2 \left (300 d+175 e x^2+126 f x^4\right )+16 a b^3 \left (90 c+50 d x^2+35 e x^4+27 f x^6\right )-32 b^4 x^2 \left (30 c+20 d x^2+15 e x^4+12 f x^6\right )\right )}{3840 b^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

(-(Sqrt[b]*x*Sqrt[a + b*x^2]*(-945*a^4*f + 210*a^3*b*(5*e + 3*f*x^2) - 4*a^2*b^2
*(300*d + 175*e*x^2 + 126*f*x^4) - 32*b^4*x^2*(30*c + 20*d*x^2 + 15*e*x^4 + 12*f
*x^6) + 16*a*b^3*(90*c + 50*d*x^2 + 35*e*x^4 + 27*f*x^6))) + 15*a^2*(96*b^3*c -
80*a*b^2*d + 70*a^2*b*e - 63*a^3*f)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(3840*b^
(11/2))

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Maple [A]  time = 0.032, size = 368, normalized size = 1.5 \[{\frac{c{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,acx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}c}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{d{x}^{5}}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,ad{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{2}dx}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{a}^{3}d}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{e{x}^{7}}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{7\,ae{x}^{5}}{48\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{35\,{a}^{2}e{x}^{3}}{192\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{35\,e{a}^{3}x}{128\,{b}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{35\,e{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{f{x}^{9}}{10\,b}\sqrt{b{x}^{2}+a}}-{\frac{9\,af{x}^{7}}{80\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{21\,{a}^{2}f{x}^{5}}{160\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{21\,{a}^{3}f{x}^{3}}{128\,{b}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{63\,f{a}^{4}x}{256\,{b}^{5}}\sqrt{b{x}^{2}+a}}-{\frac{63\,f{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x)

[Out]

1/4*c*x^3/b*(b*x^2+a)^(1/2)-3/8*c*a/b^2*x*(b*x^2+a)^(1/2)+3/8*c*a^2/b^(5/2)*ln(x
*b^(1/2)+(b*x^2+a)^(1/2))+1/6*d*x^5/b*(b*x^2+a)^(1/2)-5/24*d*a/b^2*x^3*(b*x^2+a)
^(1/2)+5/16*d*a^2/b^3*x*(b*x^2+a)^(1/2)-5/16*d*a^3/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a
)^(1/2))+1/8*e*x^7/b*(b*x^2+a)^(1/2)-7/48*e*a/b^2*x^5*(b*x^2+a)^(1/2)+35/192*e*a
^2/b^3*x^3*(b*x^2+a)^(1/2)-35/128*e*a^3/b^4*x*(b*x^2+a)^(1/2)+35/128*e*a^4/b^(9/
2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/10*f*x^9*(b*x^2+a)^(1/2)/b-9/80*f*a/b^2*x^7*(
b*x^2+a)^(1/2)+21/160*f*a^2/b^3*x^5*(b*x^2+a)^(1/2)-21/128*f*a^3/b^4*x^3*(b*x^2+
a)^(1/2)+63/256*f*a^4/b^5*x*(b*x^2+a)^(1/2)-63/256*f*a^5/b^(11/2)*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((f*x^6 + e*x^4 + d*x^2 + c)*x^4/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.557313, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, b^{4} f x^{9} + 48 \,{\left (10 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 8 \,{\left (80 \, b^{4} d - 70 \, a b^{3} e + 63 \, a^{2} b^{2} f\right )} x^{5} + 10 \,{\left (96 \, b^{4} c - 80 \, a b^{3} d + 70 \, a^{2} b^{2} e - 63 \, a^{3} b f\right )} x^{3} - 15 \,{\left (96 \, a b^{3} c - 80 \, a^{2} b^{2} d + 70 \, a^{3} b e - 63 \, a^{4} f\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d + 70 \, a^{4} b e - 63 \, a^{5} f\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{7680 \, b^{\frac{11}{2}}}, \frac{{\left (384 \, b^{4} f x^{9} + 48 \,{\left (10 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 8 \,{\left (80 \, b^{4} d - 70 \, a b^{3} e + 63 \, a^{2} b^{2} f\right )} x^{5} + 10 \,{\left (96 \, b^{4} c - 80 \, a b^{3} d + 70 \, a^{2} b^{2} e - 63 \, a^{3} b f\right )} x^{3} - 15 \,{\left (96 \, a b^{3} c - 80 \, a^{2} b^{2} d + 70 \, a^{3} b e - 63 \, a^{4} f\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d + 70 \, a^{4} b e - 63 \, a^{5} f\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{3840 \, \sqrt{-b} b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^4/sqrt(b*x^2 + a),x, algorithm="fricas")

[Out]

[1/7680*(2*(384*b^4*f*x^9 + 48*(10*b^4*e - 9*a*b^3*f)*x^7 + 8*(80*b^4*d - 70*a*b
^3*e + 63*a^2*b^2*f)*x^5 + 10*(96*b^4*c - 80*a*b^3*d + 70*a^2*b^2*e - 63*a^3*b*f
)*x^3 - 15*(96*a*b^3*c - 80*a^2*b^2*d + 70*a^3*b*e - 63*a^4*f)*x)*sqrt(b*x^2 + a
)*sqrt(b) - 15*(96*a^2*b^3*c - 80*a^3*b^2*d + 70*a^4*b*e - 63*a^5*f)*log(2*sqrt(
b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(11/2), 1/3840*((384*b^4*f*x^9 + 48*(
10*b^4*e - 9*a*b^3*f)*x^7 + 8*(80*b^4*d - 70*a*b^3*e + 63*a^2*b^2*f)*x^5 + 10*(9
6*b^4*c - 80*a*b^3*d + 70*a^2*b^2*e - 63*a^3*b*f)*x^3 - 15*(96*a*b^3*c - 80*a^2*
b^2*d + 70*a^3*b*e - 63*a^4*f)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 15*(96*a^2*b^3*c -
80*a^3*b^2*d + 70*a^4*b*e - 63*a^5*f)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(
-b)*b^5)]

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Sympy [A]  time = 52.5299, size = 586, normalized size = 2.39 \[ \frac{63 a^{\frac{9}{2}} f x}{256 b^{5} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{35 a^{\frac{7}{2}} e x}{128 b^{4} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{21 a^{\frac{7}{2}} f x^{3}}{256 b^{4} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{5}{2}} d x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{35 a^{\frac{5}{2}} e x^{3}}{384 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{21 a^{\frac{5}{2}} f x^{5}}{640 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{3}{2}} c x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{3}{2}} d x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{7 a^{\frac{3}{2}} e x^{5}}{192 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{\frac{3}{2}} f x^{7}}{160 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} c x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} d x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} e x^{7}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} f x^{9}}{80 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{63 a^{5} f \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{11}{2}}} + \frac{35 a^{4} e \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{9}{2}}} - \frac{5 a^{3} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{3 a^{2} c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{c x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{d x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{e x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{f x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

63*a**(9/2)*f*x/(256*b**5*sqrt(1 + b*x**2/a)) - 35*a**(7/2)*e*x/(128*b**4*sqrt(1
+ b*x**2/a)) + 21*a**(7/2)*f*x**3/(256*b**4*sqrt(1 + b*x**2/a)) + 5*a**(5/2)*d*
x/(16*b**3*sqrt(1 + b*x**2/a)) - 35*a**(5/2)*e*x**3/(384*b**3*sqrt(1 + b*x**2/a)
) - 21*a**(5/2)*f*x**5/(640*b**3*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*c*x/(8*b**2*sq
rt(1 + b*x**2/a)) + 5*a**(3/2)*d*x**3/(48*b**2*sqrt(1 + b*x**2/a)) + 7*a**(3/2)*
e*x**5/(192*b**2*sqrt(1 + b*x**2/a)) + 3*a**(3/2)*f*x**7/(160*b**2*sqrt(1 + b*x*
*2/a)) - sqrt(a)*c*x**3/(8*b*sqrt(1 + b*x**2/a)) - sqrt(a)*d*x**5/(24*b*sqrt(1 +
b*x**2/a)) - sqrt(a)*e*x**7/(48*b*sqrt(1 + b*x**2/a)) - sqrt(a)*f*x**9/(80*b*sq
rt(1 + b*x**2/a)) - 63*a**5*f*asinh(sqrt(b)*x/sqrt(a))/(256*b**(11/2)) + 35*a**4
*e*asinh(sqrt(b)*x/sqrt(a))/(128*b**(9/2)) - 5*a**3*d*asinh(sqrt(b)*x/sqrt(a))/(
16*b**(7/2)) + 3*a**2*c*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) + c*x**5/(4*sqrt(a
)*sqrt(1 + b*x**2/a)) + d*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + e*x**9/(8*sqrt(a
)*sqrt(1 + b*x**2/a)) + f*x**11/(10*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.228277, size = 302, normalized size = 1.23 \[ \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (\frac{8 \, f x^{2}}{b} - \frac{9 \, a b^{7} f - 10 \, b^{8} e}{b^{9}}\right )} x^{2} + \frac{80 \, b^{8} d + 63 \, a^{2} b^{6} f - 70 \, a b^{7} e}{b^{9}}\right )} x^{2} + \frac{5 \,{\left (96 \, b^{8} c - 80 \, a b^{7} d - 63 \, a^{3} b^{5} f + 70 \, a^{2} b^{6} e\right )}}{b^{9}}\right )} x^{2} - \frac{15 \,{\left (96 \, a b^{7} c - 80 \, a^{2} b^{6} d - 63 \, a^{4} b^{4} f + 70 \, a^{3} b^{5} e\right )}}{b^{9}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d - 63 \, a^{5} f + 70 \, a^{4} b e\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^4/sqrt(b*x^2 + a),x, algorithm="giac")

[Out]

1/3840*(2*(4*(6*(8*f*x^2/b - (9*a*b^7*f - 10*b^8*e)/b^9)*x^2 + (80*b^8*d + 63*a^
2*b^6*f - 70*a*b^7*e)/b^9)*x^2 + 5*(96*b^8*c - 80*a*b^7*d - 63*a^3*b^5*f + 70*a^
2*b^6*e)/b^9)*x^2 - 15*(96*a*b^7*c - 80*a^2*b^6*d - 63*a^4*b^4*f + 70*a^3*b^5*e)
/b^9)*sqrt(b*x^2 + a)*x - 1/256*(96*a^2*b^3*c - 80*a^3*b^2*d - 63*a^5*f + 70*a^4
*b*e)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)