Optimal. Leaf size=122 \[ -\frac {d x}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {d (5 b c-2 a d) x}{3 c^2 (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} (b c-a d)^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} (b c-a d)^{5/2}}-\frac {d x (5 b c-2 a d)}{3 c^2 \sqrt {c+d x^2} (b c-a d)^2}-\frac {d x}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 385
Rule 425
Rule 541
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx &=-\frac {d x}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {3 b c-2 a d-2 b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 c (b c-a d)}\\ &=-\frac {d x}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {d (5 b c-2 a d) x}{3 c^2 (b c-a d)^2 \sqrt {c+d x^2}}+\frac {\int \frac {3 b^2 c^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c^2 (b c-a d)^2}\\ &=-\frac {d x}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {d (5 b c-2 a d) x}{3 c^2 (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b^2 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^2}\\ &=-\frac {d x}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {d (5 b c-2 a d) x}{3 c^2 (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{(b c-a d)^2}\\ &=-\frac {d x}{3 c (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {d (5 b c-2 a d) x}{3 c^2 (b c-a d)^2 \sqrt {c+d x^2}}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 132, normalized size = 1.08 \begin {gather*} \frac {d x \left (a d \left (3 c+2 d x^2\right )-b c \left (6 c+5 d x^2\right )\right )}{3 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b^2 \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 )}{\sqrt {a} (b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1395\) vs.
\(2(104)=208\).
time = 0.10, size = 1396, normalized size = 11.44
method | result | size |
default | \(\text {Expression too large to display}\) | \(1396\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 362 vs.
\(2 (104) = 208\).
time = 1.52, size = 764, normalized size = 6.26 \begin {gather*} \left [-\frac {3 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\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 (5 \, a b^{2} c^{2} d^{2} - 7 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4}\right )} x^{3} + 3 \, {\left (2 \, a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{7} - 3 \, a^{2} b^{2} c^{6} d + 3 \, a^{3} b c^{5} d^{2} - a^{4} c^{4} d^{3} + {\left (a b^{3} c^{5} d^{2} - 3 \, a^{2} b^{2} c^{4} d^{3} + 3 \, a^{3} b c^{3} d^{4} - a^{4} c^{2} d^{5}\right )} x^{4} + 2 \, {\left (a b^{3} c^{6} d - 3 \, a^{2} b^{2} c^{5} d^{2} + 3 \, a^{3} b c^{4} d^{3} - a^{4} c^{3} d^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\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 (5 \, a b^{2} c^{2} d^{2} - 7 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4}\right )} x^{3} + 3 \, {\left (2 \, a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a b^{3} c^{7} - 3 \, a^{2} b^{2} c^{6} d + 3 \, a^{3} b c^{5} d^{2} - a^{4} c^{4} d^{3} + {\left (a b^{3} c^{5} d^{2} - 3 \, a^{2} b^{2} c^{4} d^{3} + 3 \, a^{3} b c^{3} d^{4} - a^{4} c^{2} d^{5}\right )} x^{4} + 2 \, {\left (a b^{3} c^{6} d - 3 \, a^{2} b^{2} c^{5} d^{2} + 3 \, a^{3} b c^{4} d^{3} - a^{4} c^{3} d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 321 vs.
\(2 (104) = 208\).
time = 0.76, size = 321, normalized size = 2.63 \begin {gather*} -\frac {b^{2} \sqrt {d} \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 )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\frac {{\left (5 \, b^{3} c^{3} d^{3} - 12 \, a b^{2} c^{2} d^{4} + 9 \, a^{2} b c d^{5} - 2 \, a^{3} d^{6}\right )} x^{2}}{b^{4} c^{6} d - 4 \, a b^{3} c^{5} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{3} - 4 \, a^{3} b c^{3} d^{4} + a^{4} c^{2} d^{5}} + \frac {3 \, {\left (2 \, b^{3} c^{4} d^{2} - 5 \, a b^{2} c^{3} d^{3} + 4 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )}}{b^{4} c^{6} d - 4 \, a b^{3} c^{5} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{3} - 4 \, a^{3} b c^{3} d^{4} + a^{4} c^{2} d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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