Optimal. Leaf size=168 \[ -\frac {(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} b^2}-\frac {c \sqrt {c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac {\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b x \left (a+b x^2\right )}+\frac {d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {468, 580, 523, 217, 206, 377, 205} \begin {gather*} -\frac {(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} b^2}-\frac {c \sqrt {c+d x^2} (3 b c-a d)}{2 a^2 b x}+\frac {\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b x \left (a+b x^2\right )}+\frac {d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 206
Rule 217
Rule 377
Rule 468
Rule 523
Rule 580
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{x^2 \left (a+b x^2\right )^2} \, dx &=\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x \left (a+b x^2\right )}-\frac {\int \frac {\sqrt {c+d x^2} \left (-c (3 b c-a d)-2 a d^2 x^2\right )}{x^2 \left (a+b x^2\right )} \, dx}{2 a b}\\ &=-\frac {c (3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x \left (a+b x^2\right )}-\frac {\int \frac {c \left (3 b^2 c^2-4 a b c d-a^2 d^2\right )-2 a^2 d^3 x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2 b}\\ &=-\frac {c (3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x \left (a+b x^2\right )}+\frac {d^3 \int \frac {1}{\sqrt {c+d x^2}} \, dx}{b^2}-\frac {\left ((b c-a d)^2 (3 b c+2 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^2 b^2}\\ &=-\frac {c (3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x \left (a+b x^2\right )}+\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^2}-\frac {\left ((b c-a d)^2 (3 b c+2 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^2 b^2}\\ &=-\frac {c (3 b c-a d) \sqrt {c+d x^2}}{2 a^2 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x \left (a+b x^2\right )}-\frac {(b c-a d)^{3/2} (3 b c+2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} b^2}+\frac {d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 150, normalized size = 0.89 \begin {gather*} -\frac {(b c-a d)^{3/2} (2 a d+3 b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} b^2}+\sqrt {c+d x^2} \left (-\frac {x (b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac {c^2}{a^2 x}\right )+\frac {d^{5/2} \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.75, size = 203, normalized size = 1.21 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-a^2 d^2 x^2-2 a b c^2+2 a b c d x^2-3 b^2 c^2 x^2\right )}{2 a^2 b x \left (a+b x^2\right )}+\frac {\sqrt {b c-a d} \left (-2 a^2 d^2-a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {a \sqrt {d}-b x \sqrt {c+d x^2}+b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{5/2} b^2}-\frac {d^{5/2} \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.85, size = 1184, normalized size = 7.05
result too large to display
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 [B] time = 0.02, size = 7529, normalized size = 44.82 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{2}}\,{d x} \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 {{\left (d\,x^2+c\right )}^{5/2}}{x^2\,{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{x^{2} \left (a + b x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________