3.57 \(\int \frac{A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=219 \[ \frac{(9 A b-2 a C) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{11/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{a \left (\frac{A b}{a}-C\right )+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}} \]

[Out]

-(a*((A*b)/a - C) + b*B*x)/(7*a^2*(a + b*x^2)^(7/2)) - (7*(2*A*b - a*C) + 13*b*B
*x)/(35*a^3*(a + b*x^2)^(5/2)) - (35*(3*A*b - a*C) + 87*b*B*x)/(105*a^4*(a + b*x
^2)^(3/2)) - (35*(4*A*b - a*C) + 93*b*B*x)/(35*a^5*Sqrt[a + b*x^2]) - (A*Sqrt[a
+ b*x^2])/(2*a^5*x^2) - (B*Sqrt[a + b*x^2])/(a^5*x) + ((9*A*b - 2*a*C)*ArcTanh[S
qrt[a + b*x^2]/Sqrt[a]])/(2*a^(11/2))

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Rubi [A]  time = 0.866946, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{(9 A b-2 a C) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{11/2}}-\frac{35 (4 A b-a C)+93 b B x}{35 a^5 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{2 a^5 x^2}-\frac{B \sqrt{a+b x^2}}{a^5 x}-\frac{35 (3 A b-a C)+87 b B x}{105 a^4 \left (a+b x^2\right )^{3/2}}-\frac{7 (2 A b-a C)+13 b B x}{35 a^3 \left (a+b x^2\right )^{5/2}}-\frac{-a C+A b+b B x}{7 a^2 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x + C*x^2)/(x^3*(a + b*x^2)^(9/2)),x]

[Out]

-(A*b - a*C + b*B*x)/(7*a^2*(a + b*x^2)^(7/2)) - (7*(2*A*b - a*C) + 13*b*B*x)/(3
5*a^3*(a + b*x^2)^(5/2)) - (35*(3*A*b - a*C) + 87*b*B*x)/(105*a^4*(a + b*x^2)^(3
/2)) - (35*(4*A*b - a*C) + 93*b*B*x)/(35*a^5*Sqrt[a + b*x^2]) - (A*Sqrt[a + b*x^
2])/(2*a^5*x^2) - (B*Sqrt[a + b*x^2])/(a^5*x) + ((9*A*b - 2*a*C)*ArcTanh[Sqrt[a
+ b*x^2]/Sqrt[a]])/(2*a^(11/2))

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Rubi in Sympy [A]  time = 85.8284, size = 207, normalized size = 0.95 \[ - \frac{A}{2 a x^{2} \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{128 B \sqrt{a + b x^{2}}}{35 a^{5} x} + \frac{2 B a - x \left (9 A b - 2 C a\right )}{14 a^{2} x \left (a + b x^{2}\right )^{\frac{7}{2}}} + \frac{16 B a - x \left (63 A b - 14 C a\right )}{70 a^{3} x \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{96 B a - x \left (315 A b - 70 C a\right )}{210 a^{4} x \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{384 B a - x \left (945 A b - 210 C a\right )}{210 a^{5} x \sqrt{a + b x^{2}}} + \frac{\left (9 A b - 2 C a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((C*x**2+B*x+A)/x**3/(b*x**2+a)**(9/2),x)

[Out]

-A/(2*a*x**2*(a + b*x**2)**(7/2)) - 128*B*sqrt(a + b*x**2)/(35*a**5*x) + (2*B*a
- x*(9*A*b - 2*C*a))/(14*a**2*x*(a + b*x**2)**(7/2)) + (16*B*a - x*(63*A*b - 14*
C*a))/(70*a**3*x*(a + b*x**2)**(5/2)) + (96*B*a - x*(315*A*b - 70*C*a))/(210*a**
4*x*(a + b*x**2)**(3/2)) + (384*B*a - x*(945*A*b - 210*C*a))/(210*a**5*x*sqrt(a
+ b*x**2)) + (9*A*b - 2*C*a)*atanh(sqrt(a + b*x**2)/sqrt(a))/(2*a**(11/2))

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Mathematica [A]  time = 0.492398, size = 183, normalized size = 0.84 \[ \frac{-\frac{\sqrt{a} \left (a^4 \left (105 A+210 B x-352 C x^2\right )+4 a^3 b x^2 (396 A+7 x (60 B-29 C x))+14 a^2 b^2 x^4 (261 A+10 x (24 B-5 C x))+42 a b^3 x^6 (75 A+x (64 B-5 C x))+3 b^4 x^8 (315 A+256 B x)\right )}{x^2 \left (a+b x^2\right )^{7/2}}+105 (9 A b-2 a C) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+105 \log (x) (2 a C-9 A b)}{210 a^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x + C*x^2)/(x^3*(a + b*x^2)^(9/2)),x]

[Out]

(-((Sqrt[a]*(3*b^4*x^8*(315*A + 256*B*x) + a^4*(105*A + 210*B*x - 352*C*x^2) + 4
*a^3*b*x^2*(396*A + 7*x*(60*B - 29*C*x)) + 14*a^2*b^2*x^4*(261*A + 10*x*(24*B -
5*C*x)) + 42*a*b^3*x^6*(75*A + x*(64*B - 5*C*x))))/(x^2*(a + b*x^2)^(7/2))) + 10
5*(-9*A*b + 2*a*C)*Log[x] + 105*(9*A*b - 2*a*C)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]]
)/(210*a^(11/2))

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Maple [A]  time = 0.016, size = 288, normalized size = 1.3 \[ -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{9\,Ab}{14\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{9\,Ab}{10\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{3\,Ab}{2\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{9\,Ab}{2\,{a}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{9\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{11}{2}}}}-{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{8\,bBx}{7\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{48\,bBx}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{64\,bBx}{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{128\,bBx}{35\,{a}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{C}{7\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{C}{5\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{C}{3\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{C}{{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{C\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)/x^3/(b*x^2+a)^(9/2),x)

[Out]

-1/2*A/a/x^2/(b*x^2+a)^(7/2)-9/14*A*b/a^2/(b*x^2+a)^(7/2)-9/10*A*b/a^3/(b*x^2+a)
^(5/2)-3/2*A*b/a^4/(b*x^2+a)^(3/2)-9/2*A*b/a^5/(b*x^2+a)^(1/2)+9/2*A*b/a^(11/2)*
ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-B/a/x/(b*x^2+a)^(7/2)-8/7*B*b/a^2*x/(b*x^2
+a)^(7/2)-48/35*B*b/a^3*x/(b*x^2+a)^(5/2)-64/35*B*b/a^4*x/(b*x^2+a)^(3/2)-128/35
*B*b/a^5*x/(b*x^2+a)^(1/2)+1/7*C/a/(b*x^2+a)^(7/2)+1/5*C/a^2/(b*x^2+a)^(5/2)+1/3
*C/a^3/(b*x^2+a)^(3/2)+C/a^4/(b*x^2+a)^(1/2)-C/a^(9/2)*ln((2*a+2*a^(1/2)*(b*x^2+
a)^(1/2))/x)

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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((C*x^2 + B*x + A)/((b*x^2 + a)^(9/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.37926, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (768 \, B b^{4} x^{9} + 2688 \, B a b^{3} x^{7} + 3360 \, B a^{2} b^{2} x^{5} - 105 \,{\left (2 \, C a b^{3} - 9 \, A b^{4}\right )} x^{8} + 1680 \, B a^{3} b x^{3} - 350 \,{\left (2 \, C a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} + 210 \, B a^{4} x + 105 \, A a^{4} - 406 \,{\left (2 \, C a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} - 176 \,{\left (2 \, C a^{4} - 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} + 105 \,{\left ({\left (2 \, C a b^{4} - 9 \, A b^{5}\right )} x^{10} + 4 \,{\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 6 \,{\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 4 \,{\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} +{\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{420 \,{\left (a^{5} b^{4} x^{10} + 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )} \sqrt{a}}, -\frac{{\left (768 \, B b^{4} x^{9} + 2688 \, B a b^{3} x^{7} + 3360 \, B a^{2} b^{2} x^{5} - 105 \,{\left (2 \, C a b^{3} - 9 \, A b^{4}\right )} x^{8} + 1680 \, B a^{3} b x^{3} - 350 \,{\left (2 \, C a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} + 210 \, B a^{4} x + 105 \, A a^{4} - 406 \,{\left (2 \, C a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} - 176 \,{\left (2 \, C a^{4} - 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} + 105 \,{\left ({\left (2 \, C a b^{4} - 9 \, A b^{5}\right )} x^{10} + 4 \,{\left (2 \, C a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{8} + 6 \,{\left (2 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{6} + 4 \,{\left (2 \, C a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{4} +{\left (2 \, C a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{210 \,{\left (a^{5} b^{4} x^{10} + 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)/((b*x^2 + a)^(9/2)*x^3),x, algorithm="fricas")

[Out]

[-1/420*(2*(768*B*b^4*x^9 + 2688*B*a*b^3*x^7 + 3360*B*a^2*b^2*x^5 - 105*(2*C*a*b
^3 - 9*A*b^4)*x^8 + 1680*B*a^3*b*x^3 - 350*(2*C*a^2*b^2 - 9*A*a*b^3)*x^6 + 210*B
*a^4*x + 105*A*a^4 - 406*(2*C*a^3*b - 9*A*a^2*b^2)*x^4 - 176*(2*C*a^4 - 9*A*a^3*
b)*x^2)*sqrt(b*x^2 + a)*sqrt(a) + 105*((2*C*a*b^4 - 9*A*b^5)*x^10 + 4*(2*C*a^2*b
^3 - 9*A*a*b^4)*x^8 + 6*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 4*(2*C*a^4*b - 9*A*a^3
*b^2)*x^4 + (2*C*a^5 - 9*A*a^4*b)*x^2)*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^
2 + a)*a)/x^2))/((a^5*b^4*x^10 + 4*a^6*b^3*x^8 + 6*a^7*b^2*x^6 + 4*a^8*b*x^4 + a
^9*x^2)*sqrt(a)), -1/210*((768*B*b^4*x^9 + 2688*B*a*b^3*x^7 + 3360*B*a^2*b^2*x^5
- 105*(2*C*a*b^3 - 9*A*b^4)*x^8 + 1680*B*a^3*b*x^3 - 350*(2*C*a^2*b^2 - 9*A*a*b
^3)*x^6 + 210*B*a^4*x + 105*A*a^4 - 406*(2*C*a^3*b - 9*A*a^2*b^2)*x^4 - 176*(2*C
*a^4 - 9*A*a^3*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a) + 105*((2*C*a*b^4 - 9*A*b^5)*x^1
0 + 4*(2*C*a^2*b^3 - 9*A*a*b^4)*x^8 + 6*(2*C*a^3*b^2 - 9*A*a^2*b^3)*x^6 + 4*(2*C
*a^4*b - 9*A*a^3*b^2)*x^4 + (2*C*a^5 - 9*A*a^4*b)*x^2)*arctan(sqrt(-a)/sqrt(b*x^
2 + a)))/((a^5*b^4*x^10 + 4*a^6*b^3*x^8 + 6*a^7*b^2*x^6 + 4*a^8*b*x^4 + a^9*x^2)
*sqrt(-a))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x**2+B*x+A)/x**3/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23069, size = 439, normalized size = 2. \[ -\frac{{\left ({\left ({\left ({\left (3 \,{\left ({\left (\frac{93 \, B b^{4} x}{a^{5}} - \frac{35 \,{\left (C a^{24} b^{6} - 4 \, A a^{23} b^{7}\right )}}{a^{28} b^{3}}\right )} x + \frac{308 \, B b^{3}}{a^{4}}\right )} x - \frac{35 \,{\left (10 \, C a^{25} b^{5} - 39 \, A a^{24} b^{6}\right )}}{a^{28} b^{3}}\right )} x + \frac{1050 \, B b^{2}}{a^{3}}\right )} x - \frac{14 \,{\left (29 \, C a^{26} b^{4} - 108 \, A a^{25} b^{5}\right )}}{a^{28} b^{3}}\right )} x + \frac{420 \, B b}{a^{2}}\right )} x - \frac{2 \,{\left (88 \, C a^{27} b^{3} - 291 \, A a^{26} b^{4}\right )}}{a^{28} b^{3}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{{\left (2 \, C a - 9 \, A b\right )} \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{5}} + \frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{3} A b + 2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} +{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)/((b*x^2 + a)^(9/2)*x^3),x, algorithm="giac")

[Out]

-1/105*(((((3*((93*B*b^4*x/a^5 - 35*(C*a^24*b^6 - 4*A*a^23*b^7)/(a^28*b^3))*x +
308*B*b^3/a^4)*x - 35*(10*C*a^25*b^5 - 39*A*a^24*b^6)/(a^28*b^3))*x + 1050*B*b^2
/a^3)*x - 14*(29*C*a^26*b^4 - 108*A*a^25*b^5)/(a^28*b^3))*x + 420*B*b/a^2)*x - 2
*(88*C*a^27*b^3 - 291*A*a^26*b^4)/(a^28*b^3))/(b*x^2 + a)^(7/2) + (2*C*a - 9*A*b
)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^5) + ((sqrt(b)*x -
sqrt(b*x^2 + a))^3*A*b + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b) + (sqrt(
b)*x - sqrt(b*x^2 + a))*A*a*b - 2*B*a^2*sqrt(b))/(((sqrt(b)*x - sqrt(b*x^2 + a))
^2 - a)^2*a^5)