Optimal. Leaf size=201 \[ \frac {d (3 b c+2 a d) x}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+16 a b c d-4 a^2 d^2\right ) x}{6 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {b^2 (b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {425, 541, 12,
385, 211} \begin {gather*} \frac {b^2 (b c-6 a d) \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{7/2}}+\frac {d x \left (-4 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{6 a c^2 \sqrt {c+d x^2} (b c-a d)^3}+\frac {b x}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {d x (2 a d+3 b c)}{6 a c \left (c+d x^2\right )^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 211
Rule 385
Rule 425
Rule 541
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {-b c+2 a d-4 b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{2 a (b c-a d)}\\ &=\frac {d (3 b c+2 a d) x}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {-3 b^2 c^2+12 a b c d-4 a^2 d^2-2 b d (3 b c+2 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{6 a c (b c-a d)^2}\\ &=\frac {d (3 b c+2 a d) x}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+16 a b c d-4 a^2 d^2\right ) x}{6 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\int -\frac {3 b^2 c^2 (b c-6 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a c^2 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d) x}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+16 a b c d-4 a^2 d^2\right ) x}{6 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {\left (b^2 (b c-6 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d) x}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+16 a b c d-4 a^2 d^2\right ) x}{6 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {\left (b^2 (b c-6 a d)\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d) x}{6 a c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+16 a b c d-4 a^2 d^2\right ) x}{6 a c^2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {b^2 (b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.16, size = 214, normalized size = 1.06 \begin {gather*} \frac {x \left (3 b^3 c^2 \left (c+d x^2\right )^2-2 a^3 d^3 \left (3 c+2 d x^2\right )+2 a b^2 c d^2 x^2 \left (9 c+8 d x^2\right )+2 a^2 b d^2 \left (9 c^2+5 c d x^2-2 d^2 x^4\right )\right )}{6 a c^2 (b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {b^2 (b c-6 a d) \tan ^{-1}\left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2} (b c-a d)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3488\) vs.
\(2(177)=354\).
time = 0.10, size = 3489, normalized size = 17.36
method | result | size |
default | \(\text {Expression too large to display}\) | \(3489\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 697 vs.
\(2 (177) = 354\).
time = 2.83, size = 1434, normalized size = 7.13 \begin {gather*} \left [-\frac {3 \, {\left (a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + {\left (b^{4} c^{3} d^{2} - 6 \, a b^{3} c^{2} d^{3}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 11 \, a b^{3} c^{3} d^{2} - 6 \, a^{2} b^{2} c^{2} d^{3}\right )} x^{4} + {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d - 12 \, a^{2} b^{2} c^{3} d^{2}\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (3 \, a b^{4} c^{3} d^{2} + 13 \, a^{2} b^{3} c^{2} d^{3} - 20 \, a^{3} b^{2} c d^{4} + 4 \, a^{4} b d^{5}\right )} x^{5} + 2 \, {\left (3 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} - 7 \, a^{4} b c d^{4} + 2 \, a^{5} d^{5}\right )} x^{3} + 3 \, {\left (a b^{4} c^{5} - a^{2} b^{3} c^{4} d + 6 \, a^{3} b^{2} c^{3} d^{2} - 8 \, a^{4} b c^{2} d^{3} + 2 \, a^{5} c d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a^{3} b^{4} c^{8} - 4 \, a^{4} b^{3} c^{7} d + 6 \, a^{5} b^{2} c^{6} d^{2} - 4 \, a^{6} b c^{5} d^{3} + a^{7} c^{4} d^{4} + {\left (a^{2} b^{5} c^{6} d^{2} - 4 \, a^{3} b^{4} c^{5} d^{3} + 6 \, a^{4} b^{3} c^{4} d^{4} - 4 \, a^{5} b^{2} c^{3} d^{5} + a^{6} b c^{2} d^{6}\right )} x^{6} + {\left (2 \, a^{2} b^{5} c^{7} d - 7 \, a^{3} b^{4} c^{6} d^{2} + 8 \, a^{4} b^{3} c^{5} d^{3} - 2 \, a^{5} b^{2} c^{4} d^{4} - 2 \, a^{6} b c^{3} d^{5} + a^{7} c^{2} d^{6}\right )} x^{4} + {\left (a^{2} b^{5} c^{8} - 2 \, a^{3} b^{4} c^{7} d - 2 \, a^{4} b^{3} c^{6} d^{2} + 8 \, a^{5} b^{2} c^{5} d^{3} - 7 \, a^{6} b c^{4} d^{4} + 2 \, a^{7} c^{3} d^{5}\right )} x^{2}\right )}}, \frac {3 \, {\left (a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + {\left (b^{4} c^{3} d^{2} - 6 \, a b^{3} c^{2} d^{3}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 11 \, a b^{3} c^{3} d^{2} - 6 \, a^{2} b^{2} c^{2} d^{3}\right )} x^{4} + {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d - 12 \, a^{2} b^{2} c^{3} d^{2}\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{4} c^{3} d^{2} + 13 \, a^{2} b^{3} c^{2} d^{3} - 20 \, a^{3} b^{2} c d^{4} + 4 \, a^{4} b d^{5}\right )} x^{5} + 2 \, {\left (3 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} - 7 \, a^{4} b c d^{4} + 2 \, a^{5} d^{5}\right )} x^{3} + 3 \, {\left (a b^{4} c^{5} - a^{2} b^{3} c^{4} d + 6 \, a^{3} b^{2} c^{3} d^{2} - 8 \, a^{4} b c^{2} d^{3} + 2 \, a^{5} c d^{4}\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{3} b^{4} c^{8} - 4 \, a^{4} b^{3} c^{7} d + 6 \, a^{5} b^{2} c^{6} d^{2} - 4 \, a^{6} b c^{5} d^{3} + a^{7} c^{4} d^{4} + {\left (a^{2} b^{5} c^{6} d^{2} - 4 \, a^{3} b^{4} c^{5} d^{3} + 6 \, a^{4} b^{3} c^{4} d^{4} - 4 \, a^{5} b^{2} c^{3} d^{5} + a^{6} b c^{2} d^{6}\right )} x^{6} + {\left (2 \, a^{2} b^{5} c^{7} d - 7 \, a^{3} b^{4} c^{6} d^{2} + 8 \, a^{4} b^{3} c^{5} d^{3} - 2 \, a^{5} b^{2} c^{4} d^{4} - 2 \, a^{6} b c^{3} d^{5} + a^{7} c^{2} d^{6}\right )} x^{4} + {\left (a^{2} b^{5} c^{8} - 2 \, a^{3} b^{4} c^{7} d - 2 \, a^{4} b^{3} c^{6} d^{2} + 8 \, a^{5} b^{2} c^{5} d^{3} - 7 \, a^{6} b c^{4} d^{4} + 2 \, a^{7} c^{3} d^{5}\right )} x^{2}\right )}}\right ] \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 {1}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 619 vs.
\(2 (177) = 354\).
time = 1.30, size = 619, normalized size = 3.08 \begin {gather*} \frac {{\left (\frac {2 \, {\left (4 \, b^{4} c^{4} d^{4} - 13 \, a b^{3} c^{3} d^{5} + 15 \, a^{2} b^{2} c^{2} d^{6} - 7 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} x^{2}}{b^{6} c^{8} d - 6 \, a b^{5} c^{7} d^{2} + 15 \, a^{2} b^{4} c^{6} d^{3} - 20 \, a^{3} b^{3} c^{5} d^{4} + 15 \, a^{4} b^{2} c^{4} d^{5} - 6 \, a^{5} b c^{3} d^{6} + a^{6} c^{2} d^{7}} + \frac {3 \, {\left (3 \, b^{4} c^{5} d^{3} - 10 \, a b^{3} c^{4} d^{4} + 12 \, a^{2} b^{2} c^{3} d^{5} - 6 \, a^{3} b c^{2} d^{6} + a^{4} c d^{7}\right )}}{b^{6} c^{8} d - 6 \, a b^{5} c^{7} d^{2} + 15 \, a^{2} b^{4} c^{6} d^{3} - 20 \, a^{3} b^{3} c^{5} d^{4} + 15 \, a^{4} b^{2} c^{4} d^{5} - 6 \, a^{5} b c^{3} d^{6} + a^{6} c^{2} d^{7}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (b^{3} c \sqrt {d} - 6 \, a b^{2} d^{\frac {3}{2}}\right )} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} d^{\frac {3}{2}} - b^{3} c^{2} \sqrt {d}}{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{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 = -0.00 \begin {gather*} \int \frac {1}{{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________