Optimal. Leaf size=123 \[ -\frac {3 c (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{8 d^{7/2}}+\frac {3 \sqrt {c+\frac {d}{x^2}} (5 b c-4 a d)}{8 d^3 x}-\frac {5 b c-4 a d}{4 d^2 x^3 \sqrt {c+\frac {d}{x^2}}}-\frac {b}{4 d x^5 \sqrt {c+\frac {d}{x^2}}} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {459, 335, 288, 321, 217, 206} \begin {gather*} \frac {3 \sqrt {c+\frac {d}{x^2}} (5 b c-4 a d)}{8 d^3 x}-\frac {5 b c-4 a d}{4 d^2 x^3 \sqrt {c+\frac {d}{x^2}}}-\frac {3 c (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{8 d^{7/2}}-\frac {b}{4 d x^5 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 288
Rule 321
Rule 335
Rule 459
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^6} \, dx &=-\frac {b}{4 d \sqrt {c+\frac {d}{x^2}} x^5}+\frac {(-5 b c+4 a d) \int \frac {1}{\left (c+\frac {d}{x^2}\right )^{3/2} x^6} \, dx}{4 d}\\ &=-\frac {b}{4 d \sqrt {c+\frac {d}{x^2}} x^5}-\frac {(-5 b c+4 a d) \operatorname {Subst}\left (\int \frac {x^4}{\left (c+d x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{4 d}\\ &=-\frac {b}{4 d \sqrt {c+\frac {d}{x^2}} x^5}-\frac {5 b c-4 a d}{4 d^2 \sqrt {c+\frac {d}{x^2}} x^3}+\frac {(3 (5 b c-4 a d)) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{4 d^2}\\ &=-\frac {b}{4 d \sqrt {c+\frac {d}{x^2}} x^5}-\frac {5 b c-4 a d}{4 d^2 \sqrt {c+\frac {d}{x^2}} x^3}+\frac {3 (5 b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d^3 x}-\frac {(3 c (5 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{8 d^3}\\ &=-\frac {b}{4 d \sqrt {c+\frac {d}{x^2}} x^5}-\frac {5 b c-4 a d}{4 d^2 \sqrt {c+\frac {d}{x^2}} x^3}+\frac {3 (5 b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d^3 x}-\frac {(3 c (5 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {1}{\sqrt {c+\frac {d}{x^2}} x}\right )}{8 d^3}\\ &=-\frac {b}{4 d \sqrt {c+\frac {d}{x^2}} x^5}-\frac {5 b c-4 a d}{4 d^2 \sqrt {c+\frac {d}{x^2}} x^3}+\frac {3 (5 b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d^3 x}-\frac {3 c (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{8 d^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 60, normalized size = 0.49 \begin {gather*} \frac {c x^4 (5 b c-4 a d) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {c x^2}{d}+1\right )-b d^2}{4 d^3 x^5 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.25, size = 126, normalized size = 1.02 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (\frac {-12 a c d x^4-4 a d^2 x^2+15 b c^2 x^4+5 b c d x^2-2 b d^2}{8 d^3 x^4 \sqrt {c x^2+d}}-\frac {3 \left (5 b c^2-4 a c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{8 d^{7/2}}\right )}{\sqrt {c x^2+d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 314, normalized size = 2.55 \begin {gather*} \left [-\frac {3 \, {\left ({\left (5 \, b c^{3} - 4 \, a c^{2} d\right )} x^{5} + {\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{3}\right )} \sqrt {d} \log \left (-\frac {c x^{2} + 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (3 \, {\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{4} - 2 \, b d^{3} + {\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, {\left (c d^{4} x^{5} + d^{5} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b c^{3} - 4 \, a c^{2} d\right )} x^{5} + {\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{3}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (3 \, {\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{4} - 2 \, b d^{3} + {\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, {\left (c d^{4} x^{5} + d^{5} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 157, normalized size = 1.28 \begin {gather*} \frac {\left (c \,x^{2}+d \right ) \left (12 \sqrt {c \,x^{2}+d}\, a c \,d^{2} x^{4} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )-15 \sqrt {c \,x^{2}+d}\, b \,c^{2} d \,x^{4} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )-12 a c \,d^{\frac {5}{2}} x^{4}+15 b \,c^{2} d^{\frac {3}{2}} x^{4}-4 a \,d^{\frac {7}{2}} x^{2}+5 b c \,d^{\frac {5}{2}} x^{2}-2 b \,d^{\frac {7}{2}}\right )}{8 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{\frac {9}{2}} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.28, size = 243, normalized size = 1.98 \begin {gather*} \frac {1}{16} \, b {\left (\frac {2 \, {\left (15 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} c^{2} x^{4} - 25 \, {\left (c + \frac {d}{x^{2}}\right )} c^{2} d x^{2} + 8 \, c^{2} d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} x^{5} - 2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{4} x^{3} + \sqrt {c + \frac {d}{x^{2}}} d^{5} x} + \frac {15 \, c^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {7}{2}}}\right )} - \frac {1}{4} \, a {\left (\frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )} c x^{2} - 2 \, c d\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} x^{3} - \sqrt {c + \frac {d}{x^{2}}} d^{3} x} + \frac {3 \, c \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {5}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+\frac {b}{x^2}}{x^6\,{\left (c+\frac {d}{x^2}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 29.81, size = 180, normalized size = 1.46 \begin {gather*} a \left (- \frac {3 \sqrt {c}}{2 d^{2} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {3 c \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2 d^{\frac {5}{2}}} - \frac {1}{2 \sqrt {c} d x^{3} \sqrt {1 + \frac {d}{c x^{2}}}}\right ) + b \left (\frac {15 c^{\frac {3}{2}}}{8 d^{3} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {5 \sqrt {c}}{8 d^{2} x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {15 c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{8 d^{\frac {7}{2}}} - \frac {1}{4 \sqrt {c} d x^{5} \sqrt {1 + \frac {d}{c x^{2}}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________