3.635 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=222 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac{\left (c+d x^2\right )^{5/2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{16 c^2 x^2}+\frac{5 d \left (c+d x^2\right )^{3/2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{48 c^2}+\frac{5 d \sqrt{c+d x^2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{16 c}-\frac{5 d \left (a d (a d+12 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 \sqrt{c}}-\frac{a \left (c+d x^2\right )^{7/2} (a d+12 b c)}{24 c^2 x^4} \]

[Out]

(5*d*(8*b^2*c^2 + a*d*(12*b*c + a*d))*Sqrt[c + d*x^2])/(16*c) + (5*d*(8*b^2*c^2
+ a*d*(12*b*c + a*d))*(c + d*x^2)^(3/2))/(48*c^2) - ((8*b^2*c^2 + a*d*(12*b*c +
a*d))*(c + d*x^2)^(5/2))/(16*c^2*x^2) - (a^2*(c + d*x^2)^(7/2))/(6*c*x^6) - (a*(
12*b*c + a*d)*(c + d*x^2)^(7/2))/(24*c^2*x^4) - (5*d*(8*b^2*c^2 + a*d*(12*b*c +
a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*Sqrt[c])

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Rubi [A]  time = 0.619153, antiderivative size = 219, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac{\left (c+d x^2\right )^{5/2} \left (\frac{a d (a d+12 b c)}{c^2}+8 b^2\right )}{16 x^2}+\frac{5 d \left (c+d x^2\right )^{3/2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{48 c^2}+\frac{5 d \sqrt{c+d x^2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{16 c}-\frac{5 d \left (a d (a d+12 b c)+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 \sqrt{c}}-\frac{a \left (c+d x^2\right )^{7/2} (a d+12 b c)}{24 c^2 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^7,x]

[Out]

(5*d*(8*b^2*c^2 + a*d*(12*b*c + a*d))*Sqrt[c + d*x^2])/(16*c) + (5*d*(8*b^2*c^2
+ a*d*(12*b*c + a*d))*(c + d*x^2)^(3/2))/(48*c^2) - ((8*b^2 + (a*d*(12*b*c + a*d
))/c^2)*(c + d*x^2)^(5/2))/(16*x^2) - (a^2*(c + d*x^2)^(7/2))/(6*c*x^6) - (a*(12
*b*c + a*d)*(c + d*x^2)^(7/2))/(24*c^2*x^4) - (5*d*(8*b^2*c^2 + a*d*(12*b*c + a*
d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*Sqrt[c])

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Rubi in Sympy [A]  time = 38.297, size = 211, normalized size = 0.95 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{6 c x^{6}} - \frac{a \left (c + d x^{2}\right )^{\frac{7}{2}} \left (a d + 12 b c\right )}{24 c^{2} x^{4}} + \frac{5 d \sqrt{c + d x^{2}} \left (a d \left (a d + 12 b c\right ) + 8 b^{2} c^{2}\right )}{16 c} + \frac{5 d \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d \left (a d + 12 b c\right ) + 8 b^{2} c^{2}\right )}{48 c^{2}} - \frac{\left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d \left (a d + 12 b c\right ) + 8 b^{2} c^{2}\right )}{16 c^{2} x^{2}} - \frac{5 d \left (a d \left (a d + 12 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{16 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x**7,x)

[Out]

-a**2*(c + d*x**2)**(7/2)/(6*c*x**6) - a*(c + d*x**2)**(7/2)*(a*d + 12*b*c)/(24*
c**2*x**4) + 5*d*sqrt(c + d*x**2)*(a*d*(a*d + 12*b*c) + 8*b**2*c**2)/(16*c) + 5*
d*(c + d*x**2)**(3/2)*(a*d*(a*d + 12*b*c) + 8*b**2*c**2)/(48*c**2) - (c + d*x**2
)**(5/2)*(a*d*(a*d + 12*b*c) + 8*b**2*c**2)/(16*c**2*x**2) - 5*d*(a*d*(a*d + 12*
b*c) + 8*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(16*sqrt(c))

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Mathematica [A]  time = 0.404278, size = 186, normalized size = 0.84 \[ \frac{1}{48} \left (-\frac{15 d \left (a^2 d^2+12 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{\sqrt{c}}-\frac{\sqrt{c+d x^2} \left (a^2 \left (8 c^2+26 c d x^2+33 d^2 x^4\right )+12 a b x^2 \left (2 c^2+9 c d x^2-8 d^2 x^4\right )-8 b^2 x^4 \left (-3 c^2+14 c d x^2+2 d^2 x^4\right )\right )}{x^6}+\frac{15 d \log (x) \left (a^2 d^2+12 a b c d+8 b^2 c^2\right )}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^7,x]

[Out]

(-((Sqrt[c + d*x^2]*(12*a*b*x^2*(2*c^2 + 9*c*d*x^2 - 8*d^2*x^4) - 8*b^2*x^4*(-3*
c^2 + 14*c*d*x^2 + 2*d^2*x^4) + a^2*(8*c^2 + 26*c*d*x^2 + 33*d^2*x^4)))/x^6) + (
15*d*(8*b^2*c^2 + 12*a*b*c*d + a^2*d^2)*Log[x])/Sqrt[c] - (15*d*(8*b^2*c^2 + 12*
a*b*c*d + a^2*d^2)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/Sqrt[c])/48

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Maple [A]  time = 0.023, size = 387, normalized size = 1.7 \[ -{\frac{{a}^{2}}{6\,c{x}^{6}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}d}{24\,{c}^{2}{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}{d}^{2}}{16\,{c}^{3}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}{d}^{3}}{16\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}{d}^{3}}{48\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{5\,{a}^{2}{d}^{3}}{16\,c}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}d}{2\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{b}^{2}d}{6} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}d}{2}{c}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) }+{\frac{5\,{b}^{2}dc}{2}\sqrt{d{x}^{2}+c}}-{\frac{ab}{2\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,abd}{4\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{3\,ab{d}^{2}}{4\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,ab{d}^{2}}{4\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{15\,ab{d}^{2}}{4}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) }+{\frac{15\,ab{d}^{2}}{4}\sqrt{d{x}^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^7,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278932, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (8 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) + 2 \,{\left (16 \, b^{2} d^{2} x^{8} + 16 \,{\left (7 \, b^{2} c d + 6 \, a b d^{2}\right )} x^{6} - 3 \,{\left (8 \, b^{2} c^{2} + 36 \, a b c d + 11 \, a^{2} d^{2}\right )} x^{4} - 8 \, a^{2} c^{2} - 2 \,{\left (12 \, a b c^{2} + 13 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{96 \, \sqrt{c} x^{6}}, -\frac{15 \,{\left (8 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (16 \, b^{2} d^{2} x^{8} + 16 \,{\left (7 \, b^{2} c d + 6 \, a b d^{2}\right )} x^{6} - 3 \,{\left (8 \, b^{2} c^{2} + 36 \, a b c d + 11 \, a^{2} d^{2}\right )} x^{4} - 8 \, a^{2} c^{2} - 2 \,{\left (12 \, a b c^{2} + 13 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{48 \, \sqrt{-c} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^7,x, algorithm="fricas")

[Out]

[1/96*(15*(8*b^2*c^2*d + 12*a*b*c*d^2 + a^2*d^3)*x^6*log(-((d*x^2 + 2*c)*sqrt(c)
- 2*sqrt(d*x^2 + c)*c)/x^2) + 2*(16*b^2*d^2*x^8 + 16*(7*b^2*c*d + 6*a*b*d^2)*x^
6 - 3*(8*b^2*c^2 + 36*a*b*c*d + 11*a^2*d^2)*x^4 - 8*a^2*c^2 - 2*(12*a*b*c^2 + 13
*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(c))/(sqrt(c)*x^6), -1/48*(15*(8*b^2*c^2*d +
12*a*b*c*d^2 + a^2*d^3)*x^6*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - (16*b^2*d^2*x^8 +
16*(7*b^2*c*d + 6*a*b*d^2)*x^6 - 3*(8*b^2*c^2 + 36*a*b*c*d + 11*a^2*d^2)*x^4 -
8*a^2*c^2 - 2*(12*a*b*c^2 + 13*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(-c))/(sqrt(-c)
*x^6)]

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Sympy [A]  time = 166.573, size = 468, normalized size = 2.11 \[ - \frac{a^{2} c^{3}}{6 \sqrt{d} x^{7} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{17 a^{2} c^{2} \sqrt{d}}{24 x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{35 a^{2} c d^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} d^{\frac{5}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} - \frac{3 a^{2} d^{\frac{5}{2}}}{16 x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{5 a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{16 \sqrt{c}} - \frac{15 a b \sqrt{c} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{4} - \frac{a b c^{3}}{2 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b c^{2} \sqrt{d}}{4 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{2 a b c d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{x} + \frac{7 a b c d^{\frac{3}{2}}}{4 x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a b d^{\frac{5}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} - \frac{5 b^{2} c^{\frac{3}{2}} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2} - \frac{b^{2} c^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} + \frac{2 b^{2} c^{2} \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 b^{2} c d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + b^{2} d^{2} \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x**7,x)

[Out]

-a**2*c**3/(6*sqrt(d)*x**7*sqrt(c/(d*x**2) + 1)) - 17*a**2*c**2*sqrt(d)/(24*x**5
*sqrt(c/(d*x**2) + 1)) - 35*a**2*c*d**(3/2)/(48*x**3*sqrt(c/(d*x**2) + 1)) - a**
2*d**(5/2)*sqrt(c/(d*x**2) + 1)/(2*x) - 3*a**2*d**(5/2)/(16*x*sqrt(c/(d*x**2) +
1)) - 5*a**2*d**3*asinh(sqrt(c)/(sqrt(d)*x))/(16*sqrt(c)) - 15*a*b*sqrt(c)*d**2*
asinh(sqrt(c)/(sqrt(d)*x))/4 - a*b*c**3/(2*sqrt(d)*x**5*sqrt(c/(d*x**2) + 1)) -
3*a*b*c**2*sqrt(d)/(4*x**3*sqrt(c/(d*x**2) + 1)) - 2*a*b*c*d**(3/2)*sqrt(c/(d*x*
*2) + 1)/x + 7*a*b*c*d**(3/2)/(4*x*sqrt(c/(d*x**2) + 1)) + 2*a*b*d**(5/2)*x/sqrt
(c/(d*x**2) + 1) - 5*b**2*c**(3/2)*d*asinh(sqrt(c)/(sqrt(d)*x))/2 - b**2*c**2*sq
rt(d)*sqrt(c/(d*x**2) + 1)/(2*x) + 2*b**2*c**2*sqrt(d)/(x*sqrt(c/(d*x**2) + 1))
+ 2*b**2*c*d**(3/2)*x/sqrt(c/(d*x**2) + 1) + b**2*d**2*Piecewise((sqrt(c)*x**2/2
, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d), True))

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GIAC/XCAS [A]  time = 0.25007, size = 386, normalized size = 1.74 \[ \frac{16 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} d^{2} + 96 \, \sqrt{d x^{2} + c} b^{2} c d^{2} + 96 \, \sqrt{d x^{2} + c} a b d^{3} + \frac{15 \,{\left (8 \, b^{2} c^{2} d^{2} + 12 \, a b c d^{3} + a^{2} d^{4}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d^{2} - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d^{2} + 108 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{3} - 192 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c^{2} d^{3} + 84 \, \sqrt{d x^{2} + c} a b c^{3} d^{3} + 33 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{4} - 40 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{4} + 15 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{4}}{d^{3} x^{6}}}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^7,x, algorithm="giac")

[Out]

1/48*(16*(d*x^2 + c)^(3/2)*b^2*d^2 + 96*sqrt(d*x^2 + c)*b^2*c*d^2 + 96*sqrt(d*x^
2 + c)*a*b*d^3 + 15*(8*b^2*c^2*d^2 + 12*a*b*c*d^3 + a^2*d^4)*arctan(sqrt(d*x^2 +
c)/sqrt(-c))/sqrt(-c) - (24*(d*x^2 + c)^(5/2)*b^2*c^2*d^2 - 48*(d*x^2 + c)^(3/2
)*b^2*c^3*d^2 + 24*sqrt(d*x^2 + c)*b^2*c^4*d^2 + 108*(d*x^2 + c)^(5/2)*a*b*c*d^3
- 192*(d*x^2 + c)^(3/2)*a*b*c^2*d^3 + 84*sqrt(d*x^2 + c)*a*b*c^3*d^3 + 33*(d*x^
2 + c)^(5/2)*a^2*d^4 - 40*(d*x^2 + c)^(3/2)*a^2*c*d^4 + 15*sqrt(d*x^2 + c)*a^2*c
^2*d^4)/(d^3*x^6))/d