3.351 \(\int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=115 \[ -\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{5 B c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{5 B c^2 \sqrt{a+c x^2}}{16 x^2}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4} \]

[Out]

(-5*B*c^2*Sqrt[a + c*x^2])/(16*x^2) - (5*B*c*(a + c*x^2)^(3/2))/(24*x^4) - (B*(a
+ c*x^2)^(5/2))/(6*x^6) - (A*(a + c*x^2)^(7/2))/(7*a*x^7) - (5*B*c^3*ArcTanh[Sq
rt[a + c*x^2]/Sqrt[a]])/(16*Sqrt[a])

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Rubi [A]  time = 0.185399, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{5 B c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{5 B c^2 \sqrt{a+c x^2}}{16 x^2}-\frac{B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac{5 B c \left (a+c x^2\right )^{3/2}}{24 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(5/2))/x^8,x]

[Out]

(-5*B*c^2*Sqrt[a + c*x^2])/(16*x^2) - (5*B*c*(a + c*x^2)^(3/2))/(24*x^4) - (B*(a
+ c*x^2)^(5/2))/(6*x^6) - (A*(a + c*x^2)^(7/2))/(7*a*x^7) - (5*B*c^3*ArcTanh[Sq
rt[a + c*x^2]/Sqrt[a]])/(16*Sqrt[a])

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Rubi in Sympy [A]  time = 17.0101, size = 109, normalized size = 0.95 \[ - \frac{A \left (a + c x^{2}\right )^{\frac{7}{2}}}{7 a x^{7}} - \frac{5 B c^{2} \sqrt{a + c x^{2}}}{16 x^{2}} - \frac{5 B c \left (a + c x^{2}\right )^{\frac{3}{2}}}{24 x^{4}} - \frac{B \left (a + c x^{2}\right )^{\frac{5}{2}}}{6 x^{6}} - \frac{5 B c^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{16 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/x**8,x)

[Out]

-A*(a + c*x**2)**(7/2)/(7*a*x**7) - 5*B*c**2*sqrt(a + c*x**2)/(16*x**2) - 5*B*c*
(a + c*x**2)**(3/2)/(24*x**4) - B*(a + c*x**2)**(5/2)/(6*x**6) - 5*B*c**3*atanh(
sqrt(a + c*x**2)/sqrt(a))/(16*sqrt(a))

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Mathematica [A]  time = 0.405559, size = 124, normalized size = 1.08 \[ -\frac{\frac{\sqrt{a+c x^2} \left (8 a^3 (6 A+7 B x)+2 a^2 c x^2 (72 A+91 B x)+3 a c^2 x^4 (48 A+77 B x)+48 A c^3 x^6\right )}{x^7}+105 \sqrt{a} B c^3 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-105 \sqrt{a} B c^3 \log (x)}{336 a} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^8,x]

[Out]

-((Sqrt[a + c*x^2]*(48*A*c^3*x^6 + 8*a^3*(6*A + 7*B*x) + 3*a*c^2*x^4*(48*A + 77*
B*x) + 2*a^2*c*x^2*(72*A + 91*B*x)))/x^7 - 105*Sqrt[a]*B*c^3*Log[x] + 105*Sqrt[a
]*B*c^3*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(336*a)

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Maple [A]  time = 0.023, size = 164, normalized size = 1.4 \[ -{\frac{A}{7\,a{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B}{6\,a{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Bc}{24\,{a}^{2}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{c}^{2}}{16\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{c}^{3}}{16\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{c}^{3}}{48\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{c}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{5\,B{c}^{3}}{16\,a}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(5/2)/x^8,x)

[Out]

-1/7*A*(c*x^2+a)^(7/2)/a/x^7-1/6*B/a/x^6*(c*x^2+a)^(7/2)-1/24*B/a^2*c/x^4*(c*x^2
+a)^(7/2)-1/16*B/a^3*c^2/x^2*(c*x^2+a)^(7/2)+1/16*B/a^3*c^3*(c*x^2+a)^(5/2)+5/48
*B/a^2*c^3*(c*x^2+a)^(3/2)-5/16*B/a^(1/2)*c^3*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))
/x)+5/16*B/a*c^3*(c*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((c*x^2 + a)^(5/2)*(B*x + A)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.362205, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a c^{3} x^{7} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{672 \, a^{\frac{3}{2}} x^{7}}, -\frac{105 \, B a c^{3} x^{7} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{336 \, \sqrt{-a} a x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^8,x, algorithm="fricas")

[Out]

[1/672*(105*B*a*c^3*x^7*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2)
- 2*(48*A*c^3*x^6 + 231*B*a*c^2*x^5 + 144*A*a*c^2*x^4 + 182*B*a^2*c*x^3 + 144*A*
a^2*c*x^2 + 56*B*a^3*x + 48*A*a^3)*sqrt(c*x^2 + a)*sqrt(a))/(a^(3/2)*x^7), -1/33
6*(105*B*a*c^3*x^7*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (48*A*c^3*x^6 + 231*B*a*c^
2*x^5 + 144*A*a*c^2*x^4 + 182*B*a^2*c*x^3 + 144*A*a^2*c*x^2 + 56*B*a^3*x + 48*A*
a^3)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^7)]

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Sympy [A]  time = 48.0918, size = 605, normalized size = 5.26 \[ - \frac{15 A a^{7} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{33 A a^{6} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{17 A a^{5} c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{3 A a^{4} c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{12 A a^{3} c^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{8 A a^{2} c^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{2 A a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{7 A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{A c^{\frac{7}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a} - \frac{B a^{3}}{6 \sqrt{c} x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{17 B a^{2} \sqrt{c}}{24 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{35 B a c^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{3 B c^{\frac{5}{2}}}{16 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 B c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{16 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(5/2)/x**8,x)

[Out]

-15*A*a**7*c**(9/2)*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**
8 + 105*a**3*c**6*x**10) - 33*A*a**6*c**(11/2)*x**2*sqrt(a/(c*x**2) + 1)/(105*a*
*5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 17*A*a**5*c**(13/2)*x
**4*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**
6*x**10) - 3*A*a**4*c**(15/2)*x**6*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 21
0*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 12*A*a**3*c**(17/2)*x**8*sqrt(a/(c*x**
2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 8*A*a*
*2*c**(19/2)*x**10*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8
+ 105*a**3*c**6*x**10) - 2*A*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 7*A*c**
(5/2)*sqrt(a/(c*x**2) + 1)/(15*x**2) - A*c**(7/2)*sqrt(a/(c*x**2) + 1)/(15*a) -
B*a**3/(6*sqrt(c)*x**7*sqrt(a/(c*x**2) + 1)) - 17*B*a**2*sqrt(c)/(24*x**5*sqrt(a
/(c*x**2) + 1)) - 35*B*a*c**(3/2)/(48*x**3*sqrt(a/(c*x**2) + 1)) - B*c**(5/2)*sq
rt(a/(c*x**2) + 1)/(2*x) - 3*B*c**(5/2)/(16*x*sqrt(a/(c*x**2) + 1)) - 5*B*c**3*a
sinh(sqrt(a)/(sqrt(c)*x))/(16*sqrt(a))

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GIAC/XCAS [A]  time = 0.284215, size = 427, normalized size = 3.71 \[ \frac{5 \, B c^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a}} + \frac{231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} B c^{3} + 336 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{12} A c^{\frac{7}{2}} - 196 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} B a c^{3} + 595 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B a^{2} c^{3} + 1680 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A a^{2} c^{\frac{7}{2}} - 595 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B a^{4} c^{3} + 1008 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{4} c^{\frac{7}{2}} + 196 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{5} c^{3} - 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{6} c^{3} + 48 \, A a^{6} c^{\frac{7}{2}}}{168 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^8,x, algorithm="giac")

[Out]

5/8*B*c^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) + 1/168*(231*
(sqrt(c)*x - sqrt(c*x^2 + a))^13*B*c^3 + 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*A*
c^(7/2) - 196*(sqrt(c)*x - sqrt(c*x^2 + a))^11*B*a*c^3 + 595*(sqrt(c)*x - sqrt(c
*x^2 + a))^9*B*a^2*c^3 + 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A*a^2*c^(7/2) - 59
5*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*a^4*c^3 + 1008*(sqrt(c)*x - sqrt(c*x^2 + a))
^4*A*a^4*c^(7/2) + 196*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^5*c^3 - 231*(sqrt(c)*
x - sqrt(c*x^2 + a))*B*a^6*c^3 + 48*A*a^6*c^(7/2))/((sqrt(c)*x - sqrt(c*x^2 + a)
)^2 - a)^7