3.337 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=124 \[ \frac{A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{A c^2 \sqrt{a+c x^2}}{16 a x^2}-\frac{A \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac{A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac{B \left (a+c x^2\right )^{5/2}}{5 a x^5} \]

[Out]

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

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Rubi [A]  time = 0.220714, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{A c^2 \sqrt{a+c x^2}}{16 a x^2}-\frac{A \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac{A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac{B \left (a+c x^2\right )^{5/2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(3/2))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 20.8523, size = 107, normalized size = 0.86 \[ \frac{A c^{2} \sqrt{a + c x^{2}}}{16 a x^{2}} + \frac{A c \left (a + c x^{2}\right )^{\frac{3}{2}}}{24 a x^{4}} - \frac{A \left (a + c x^{2}\right )^{\frac{5}{2}}}{6 a x^{6}} + \frac{A c^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{3}{2}}} - \frac{B \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.157184, size = 116, normalized size = 0.94 \[ \frac{-\sqrt{a} \sqrt{a+c x^2} \left (8 a^2 (5 A+6 B x)+2 a c x^2 (35 A+48 B x)+3 c^2 x^4 (5 A+16 B x)\right )+15 A c^3 x^6 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-15 A c^3 x^6 \log (x)}{240 a^{3/2} x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(3/2))/x^7,x]

[Out]

(-(Sqrt[a]*Sqrt[a + c*x^2]*(8*a^2*(5*A + 6*B*x) + 3*c^2*x^4*(5*A + 16*B*x) + 2*a
*c*x^2*(35*A + 48*B*x))) - 15*A*c^3*x^6*Log[x] + 15*A*c^3*x^6*Log[a + Sqrt[a]*Sq
rt[a + c*x^2]])/(240*a^(3/2)*x^6)

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Maple [A]  time = 0.018, size = 146, normalized size = 1.2 \[ -{\frac{A}{6\,a{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ac}{24\,{a}^{2}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{c}^{2}}{48\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{A{c}^{3}}{48\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{c}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A{c}^{3}}{16\,{a}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{B}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(3/2)/x^7,x)

[Out]

-1/6*A*(c*x^2+a)^(5/2)/a/x^6+1/24*A/a^2*c/x^4*(c*x^2+a)^(5/2)+1/48*A/a^3*c^2/x^2
*(c*x^2+a)^(5/2)-1/48*A/a^3*c^3*(c*x^2+a)^(3/2)+1/16*A/a^(3/2)*c^3*ln((2*a+2*a^(
1/2)*(c*x^2+a)^(1/2))/x)-1/16*A/a^2*c^3*(c*x^2+a)^(1/2)-1/5*B*(c*x^2+a)^(5/2)/a/
x^5

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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)^(3/2)*(B*x + A)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.372691, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A c^{3} x^{6} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (48 \, B c^{2} x^{5} + 15 \, A c^{2} x^{4} + 96 \, B a c x^{3} + 70 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{480 \, a^{\frac{3}{2}} x^{6}}, \frac{15 \, A c^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (48 \, B c^{2} x^{5} + 15 \, A c^{2} x^{4} + 96 \, B a c x^{3} + 70 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{240 \, \sqrt{-a} a x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

[1/480*(15*A*c^3*x^6*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2) - 2
*(48*B*c^2*x^5 + 15*A*c^2*x^4 + 96*B*a*c*x^3 + 70*A*a*c*x^2 + 48*B*a^2*x + 40*A*
a^2)*sqrt(c*x^2 + a)*sqrt(a))/(a^(3/2)*x^6), 1/240*(15*A*c^3*x^6*arctan(sqrt(-a)
/sqrt(c*x^2 + a)) - (48*B*c^2*x^5 + 15*A*c^2*x^4 + 96*B*a*c*x^3 + 70*A*a*c*x^2 +
48*B*a^2*x + 40*A*a^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^6)]

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Sympy [A]  time = 36.4498, size = 201, normalized size = 1.62 \[ - \frac{A a^{2}}{6 \sqrt{c} x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{11 A a \sqrt{c}}{24 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{17 A c^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{\frac{5}{2}}}{16 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{16 a^{\frac{3}{2}}} - \frac{B a \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{2 B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{2}} - \frac{B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(3/2)/x**7,x)

[Out]

-A*a**2/(6*sqrt(c)*x**7*sqrt(a/(c*x**2) + 1)) - 11*A*a*sqrt(c)/(24*x**5*sqrt(a/(
c*x**2) + 1)) - 17*A*c**(3/2)/(48*x**3*sqrt(a/(c*x**2) + 1)) - A*c**(5/2)/(16*a*
x*sqrt(a/(c*x**2) + 1)) + A*c**3*asinh(sqrt(a)/(sqrt(c)*x))/(16*a**(3/2)) - B*a*
sqrt(c)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 2*B*c**(3/2)*sqrt(a/(c*x**2) + 1)/(5*x**
2) - B*c**(5/2)*sqrt(a/(c*x**2) + 1)/(5*a)

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GIAC/XCAS [A]  time = 0.279208, size = 512, normalized size = 4.13 \[ -\frac{A c^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a} + \frac{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} A c^{3} + 240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} B a c^{\frac{5}{2}} + 235 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} A a c^{3} - 240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} B a^{2} c^{\frac{5}{2}} + 390 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a^{2} c^{3} + 480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{3} c^{\frac{5}{2}} + 390 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{3} c^{3} - 480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{4} c^{\frac{5}{2}} + 235 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{4} c^{3} + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{5} c^{\frac{5}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{5} c^{3} - 48 \, B a^{6} c^{\frac{5}{2}}}{120 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{6} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^7,x, algorithm="giac")

[Out]

-1/8*A*c^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/120*
(15*(sqrt(c)*x - sqrt(c*x^2 + a))^11*A*c^3 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^1
0*B*a*c^(5/2) + 235*(sqrt(c)*x - sqrt(c*x^2 + a))^9*A*a*c^3 - 240*(sqrt(c)*x - s
qrt(c*x^2 + a))^8*B*a^2*c^(5/2) + 390*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*a^2*c^3
+ 480*(sqrt(c)*x - sqrt(c*x^2 + a))^6*B*a^3*c^(5/2) + 390*(sqrt(c)*x - sqrt(c*x^
2 + a))^5*A*a^3*c^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^4*B*a^4*c^(5/2) + 235*(s
qrt(c)*x - sqrt(c*x^2 + a))^3*A*a^4*c^3 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a
^5*c^(5/2) + 15*(sqrt(c)*x - sqrt(c*x^2 + a))*A*a^5*c^3 - 48*B*a^6*c^(5/2))/(((s
qrt(c)*x - sqrt(c*x^2 + a))^2 - a)^6*a)