Optimal. Leaf size=100 \[ -\frac {c}{4 d (b c-a d) \left (c+d x^2\right )^2}-\frac {a}{2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {a b \log \left (a+b x^2\right )}{2 (b c-a d)^3}+\frac {a b \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78}
\begin {gather*} -\frac {a}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {c}{4 d \left (c+d x^2\right )^2 (b c-a d)}-\frac {a b \log \left (a+b x^2\right )}{2 (b c-a d)^3}+\frac {a b \log \left (c+d x^2\right )}{2 (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x) (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a b^2}{(b c-a d)^3 (a+b x)}+\frac {c}{(b c-a d) (c+d x)^3}+\frac {a d}{(-b c+a d)^2 (c+d x)^2}-\frac {a b d}{(-b c+a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {c}{4 d (b c-a d) \left (c+d x^2\right )^2}-\frac {a}{2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {a b \log \left (a+b x^2\right )}{2 (b c-a d)^3}+\frac {a b \log \left (c+d x^2\right )}{2 (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 77, normalized size = 0.77 \begin {gather*} \frac {\frac {(-b c+a d) \left (b c^2+a d \left (c+2 d x^2\right )\right )}{d \left (c+d x^2\right )^2}-2 a b \log \left (a+b x^2\right )+2 a b \log \left (c+d x^2\right )}{4 (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 105, normalized size = 1.05
method | result | size |
default | \(\frac {a b \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{3}}+\frac {-\frac {\left (a d -b c \right ) a}{d \,x^{2}+c}+\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}-a b \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{3}}\) | \(105\) |
risch | \(\frac {-\frac {a d \,x^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {c \left (a d +b c \right )}{4 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d \,x^{2}+c \right )^{2}}-\frac {a b \ln \left (-d \,x^{2}-c \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {a b \ln \left (b \,x^{2}+a \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\) | \(182\) |
norman | \(\frac {-\frac {a d \,x^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-a \,d^{2}-b c d \right ) c}{4 d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d \,x^{2}+c \right )^{2}}+\frac {a b \ln \left (b \,x^{2}+a \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}-\frac {a b \ln \left (d \,x^{2}+c \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 217 vs.
\(2 (92) = 184\).
time = 0.31, size = 217, normalized size = 2.17 \begin {gather*} -\frac {a b \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {a b \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {2 \, a d^{2} x^{2} + b c^{2} + a c d}{4 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\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. 256 vs.
\(2 (92) = 184\).
time = 1.00, size = 256, normalized size = 2.56 \begin {gather*} -\frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b d^{3} x^{4} + 2 \, a b c d^{2} x^{2} + a b c^{2} d\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (a b d^{3} x^{4} + 2 \, a b c d^{2} x^{2} + a b c^{2} d\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 411 vs.
\(2 (82) = 164\).
time = 2.70, size = 411, normalized size = 4.11 \begin {gather*} - \frac {a b \log {\left (x^{2} + \frac {- \frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac {a b \log {\left (x^{2} + \frac {\frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac {- a c d - 2 a d^{2} x^{2} - b c^{2}}{4 a^{2} c^{2} d^{3} - 8 a b c^{3} d^{2} + 4 b^{2} c^{4} d + x^{4} \cdot \left (4 a^{2} d^{5} - 8 a b c d^{4} + 4 b^{2} c^{2} d^{3}\right ) + x^{2} \cdot \left (8 a^{2} c d^{4} - 16 a b c^{2} d^{3} + 8 b^{2} c^{3} d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 174, normalized size = 1.74 \begin {gather*} -\frac {a b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} + \frac {a b d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} - \frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x^{2}}{4 \, {\left (d x^{2} + c\right )}^{2} {\left (b c - a d\right )}^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 343, normalized size = 3.43 \begin {gather*} -\frac {b^2\,c^3-a^2\,c\,d^2-2\,a^2\,d^3\,x^2+2\,a\,b\,c\,d^2\,x^2+a\,b\,c^2\,d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}+a\,b\,d^3\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}+a\,b\,c\,d^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,8{}\mathrm {i}}{-4\,a^3\,c^2\,d^4-8\,a^3\,c\,d^5\,x^2-4\,a^3\,d^6\,x^4+12\,a^2\,b\,c^3\,d^3+24\,a^2\,b\,c^2\,d^4\,x^2+12\,a^2\,b\,c\,d^5\,x^4-12\,a\,b^2\,c^4\,d^2-24\,a\,b^2\,c^3\,d^3\,x^2-12\,a\,b^2\,c^2\,d^4\,x^4+4\,b^3\,c^5\,d+8\,b^3\,c^4\,d^2\,x^2+4\,b^3\,c^3\,d^3\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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