Optimal. Leaf size=211 \[ -\frac {8 b^2 c^2-27 a b c d+15 a^2 d^2}{8 a c^3 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x \left (c+d x^2\right )^2}-\frac {d (9 b c-5 a d)}{8 c^2 (b c-a d)^2 x \left (c+d x^2\right )}-\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (b c-a d)^3}+\frac {d^{3/2} \left (35 b^2 c^2-42 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} (b c-a d)^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {483, 593, 597,
536, 211} \begin {gather*} -\frac {b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (b c-a d)^3}+\frac {d^{3/2} \left (15 a^2 d^2-42 a b c d+35 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} (b c-a d)^3}-\frac {15 a^2 d^2-27 a b c d+8 b^2 c^2}{8 a c^3 x (b c-a d)^2}-\frac {d (9 b c-5 a d)}{8 c^2 x \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c x \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 483
Rule 536
Rule 593
Rule 597
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=-\frac {d}{4 c (b c-a d) x \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-5 a d-5 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)}\\ &=-\frac {d}{4 c (b c-a d) x \left (c+d x^2\right )^2}-\frac {d (9 b c-5 a d)}{8 c^2 (b c-a d)^2 x \left (c+d x^2\right )}+\frac {\int \frac {8 b^2 c^2-27 a b c d+15 a^2 d^2-3 b d (9 b c-5 a d) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^2}\\ &=-\frac {\frac {8 b^2 c}{a}-27 b d+\frac {15 a d^2}{c}}{8 c^2 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x \left (c+d x^2\right )^2}-\frac {d (9 b c-5 a d)}{8 c^2 (b c-a d)^2 x \left (c+d x^2\right )}-\frac {\int \frac {8 b^3 c^3+8 a b^2 c^2 d-27 a^2 b c d^2+15 a^3 d^3+b d \left (8 b^2 c^2-27 a b c d+15 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 a c^3 (b c-a d)^2}\\ &=-\frac {\frac {8 b^2 c}{a}-27 b d+\frac {15 a d^2}{c}}{8 c^2 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x \left (c+d x^2\right )^2}-\frac {d (9 b c-5 a d)}{8 c^2 (b c-a d)^2 x \left (c+d x^2\right )}-\frac {b^4 \int \frac {1}{a+b x^2} \, dx}{a (b c-a d)^3}+\frac {\left (d^2 \left (35 b^2 c^2-42 a b c d+15 a^2 d^2\right )\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^3 (b c-a d)^3}\\ &=-\frac {\frac {8 b^2 c}{a}-27 b d+\frac {15 a d^2}{c}}{8 c^2 (b c-a d)^2 x}-\frac {d}{4 c (b c-a d) x \left (c+d x^2\right )^2}-\frac {d (9 b c-5 a d)}{8 c^2 (b c-a d)^2 x \left (c+d x^2\right )}-\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (b c-a d)^3}+\frac {d^{3/2} \left (35 b^2 c^2-42 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 172, normalized size = 0.82 \begin {gather*} \frac {1}{8} \left (-\frac {8}{a c^3 x}+\frac {2 d^2 x}{c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {d^2 (11 b c-7 a d) x}{c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac {8 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (-b c+a d)^3}+\frac {d^{3/2} \left (35 b^2 c^2-42 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 170, normalized size = 0.81
method | result | size |
default | \(\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \left (a d -b c \right )^{3} \sqrt {a b}}-\frac {d^{2} \left (\frac {\left (\frac {7}{8} a^{2} d^{3}-\frac {9}{4} a b c \,d^{2}+\frac {11}{8} b^{2} c^{2} d \right ) x^{3}+\frac {c \left (9 a^{2} d^{2}-22 a b c d +13 b^{2} c^{2}\right ) x}{8}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (15 a^{2} d^{2}-42 a b c d +35 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}}\right )}{c^{3} \left (a d -b c \right )^{3}}-\frac {1}{a \,c^{3} x}\) | \(170\) |
risch | \(\text {Expression too large to display}\) | \(2304\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 352, normalized size = 1.67 \begin {gather*} -\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\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}} + \frac {{\left (35 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 15 \, a^{2} d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} \sqrt {c d}} - \frac {8 \, b^{2} c^{4} - 16 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} - 27 \, a b c d^{3} + 15 \, a^{2} d^{4}\right )} x^{4} + {\left (16 \, b^{2} c^{3} d - 45 \, a b c^{2} d^{2} + 25 \, a^{2} c d^{3}\right )} x^{2}}{8 \, {\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{5} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{3} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x\right )}} \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. 474 vs.
\(2 (187) = 374\).
time = 2.90, size = 1991, normalized size = 9.44 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 236, normalized size = 1.12 \begin {gather*} -\frac {b^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\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}} + \frac {{\left (35 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 15 \, a^{2} d^{4}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} \sqrt {c d}} + \frac {11 \, b c d^{3} x^{3} - 7 \, a d^{4} x^{3} + 13 \, b c^{2} d^{2} x - 9 \, a c d^{3} x}{8 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {1}{a c^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 738, normalized size = 3.50 \begin {gather*} -\frac {\frac {1}{a\,c}+\frac {x^4\,\left (15\,a^2\,d^4-27\,a\,b\,c\,d^3+8\,b^2\,c^2\,d^2\right )}{8\,a\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (25\,a^2\,d^3-45\,a\,b\,c\,d^2+16\,b^2\,c^2\,d\right )}{8\,a\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2\,x+2\,c\,d\,x^3+d^2\,x^5}-\frac {\mathrm {atan}\left (\frac {b\,c^7\,x\,{\left (-a^3\,b^7\right )}^{3/2}\,64{}\mathrm {i}+a^{10}\,b\,d^7\,x\,\sqrt {-a^3\,b^7}\,225{}\mathrm {i}+a^6\,b^5\,c^4\,d^3\,x\,\sqrt {-a^3\,b^7}\,1225{}\mathrm {i}-a^7\,b^4\,c^3\,d^4\,x\,\sqrt {-a^3\,b^7}\,2940{}\mathrm {i}+a^8\,b^3\,c^2\,d^5\,x\,\sqrt {-a^3\,b^7}\,2814{}\mathrm {i}-a^9\,b^2\,c\,d^6\,x\,\sqrt {-a^3\,b^7}\,1260{}\mathrm {i}}{a^3\,b^7\,\left (2940\,a^6\,c^3\,d^4-1225\,a^5\,b\,c^4\,d^3+64\,a^2\,b^4\,c^7\right )-225\,a^{12}\,b^4\,d^7+1260\,a^{11}\,b^5\,c\,d^6-2814\,a^{10}\,b^6\,c^2\,d^5}\right )\,\sqrt {-a^3\,b^7}\,1{}\mathrm {i}}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}-\frac {\mathrm {atan}\left (\frac {a^7\,d^5\,x\,{\left (-c^7\,d^3\right )}^{3/2}\,225{}\mathrm {i}+b^7\,c^{14}\,d\,x\,\sqrt {-c^7\,d^3}\,64{}\mathrm {i}-a^4\,b^3\,c^3\,d^2\,x\,{\left (-c^7\,d^3\right )}^{3/2}\,2940{}\mathrm {i}+a^5\,b^2\,c^2\,d^3\,x\,{\left (-c^7\,d^3\right )}^{3/2}\,2814{}\mathrm {i}-a^6\,b\,c\,d^4\,x\,{\left (-c^7\,d^3\right )}^{3/2}\,1260{}\mathrm {i}+a^3\,b^4\,c^4\,d\,x\,{\left (-c^7\,d^3\right )}^{3/2}\,1225{}\mathrm {i}}{225\,a^7\,c^{11}\,d^9-1260\,a^6\,b\,c^{12}\,d^8+2814\,a^5\,b^2\,c^{13}\,d^7-2940\,a^4\,b^3\,c^{14}\,d^6+1225\,a^3\,b^4\,c^{15}\,d^5-64\,b^7\,c^{18}\,d^2}\right )\,\sqrt {-c^7\,d^3}\,\left (15\,a^2\,d^2-42\,a\,b\,c\,d+35\,b^2\,c^2\right )\,1{}\mathrm {i}}{8\,\left (-a^3\,c^7\,d^3+3\,a^2\,b\,c^8\,d^2-3\,a\,b^2\,c^9\,d+b^3\,c^{10}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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