Optimal. Leaf size=149 \[ \frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}+\frac {x \sqrt {a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {382, 378, 377, 208} \[ \frac {x \sqrt {a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 377
Rule 378
Rule 382
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^3} \, dx &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac {(a (4 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac {(a (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {(4 b c-3 a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d) \left (c+d x^2\right )}+\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.57, size = 176, normalized size = 1.18 \[ \frac {x \left (c \left (a^2 d \left (5 c+3 d x^2\right )+a b \left (-4 c^2+3 c d x^2+3 d^2 x^4\right )-2 b^2 c x^2 \left (2 c+d x^2\right )\right )+\frac {a \left (c+d x^2\right )^2 (3 a d-4 b c) \tanh ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )}{\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}}\right )}{8 c^3 \sqrt {a+b x^2} \left (c+d x^2\right )^2 (a d-b c)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.41, size = 698, normalized size = 4.68 \[ \left [\frac {{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d + {\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \, {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{3} + {\left (4 \, b^{2} c^{4} - 9 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b^{2} c^{7} - 2 \, a b c^{6} d + a^{2} c^{5} d^{2} + {\left (b^{2} c^{5} d^{2} - 2 \, a b c^{4} d^{3} + a^{2} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d + {\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \, {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (2 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{3} + {\left (4 \, b^{2} c^{4} - 9 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{2} c^{7} - 2 \, a b c^{6} d + a^{2} c^{5} d^{2} + {\left (b^{2} c^{5} d^{2} - 2 \, a b c^{4} d^{3} + a^{2} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 3.75, size = 487, normalized size = 3.27 \[ -\frac {{\left (4 \, a b^{\frac {3}{2}} c - 3 \, a^{2} \sqrt {b} d\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{8 \, \sqrt {-b^{2} c^{2} + a b c d} {\left (b c^{3} - a c^{2} d\right )}} - \frac {4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {3}{2}} c d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{3} - 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} + 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d - 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{3} - 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d + 28 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{3} - 2 \, a^{4} b^{\frac {3}{2}} c d^{2} + 3 \, a^{5} \sqrt {b} d^{3}}{4 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2} {\left (b c^{3} d - a c^{2} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 5101, normalized size = 34.23 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {b\,x^2+a}}{{\left (d\,x^2+c\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________