Integrand size = 22, antiderivative size = 115 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {(b c-a d)^3 x^2}{2 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^2 (3 b c-a d) x^6}{6 b^2}+\frac {d^3 x^8}{8 b}-\frac {a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5} \]
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Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78} \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}-\frac {a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5}+\frac {x^2 (b c-a d)^3}{2 b^4}+\frac {d^2 x^6 (3 b c-a d)}{6 b^2}+\frac {d^3 x^8}{8 b} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x (c+d x)^3}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {(b c-a d)^3}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^2}{b^2}+\frac {d^3 x^3}{b}+\frac {a (-b c+a d)^3}{b^4 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {(b c-a d)^3 x^2}{2 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^2 (3 b c-a d) x^6}{6 b^2}+\frac {d^3 x^8}{8 b}-\frac {a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {b x^2 \left (-12 a^3 d^3+6 a^2 b d^2 \left (6 c+d x^2\right )-2 a b^2 d \left (18 c^2+9 c d x^2+2 d^2 x^4\right )+3 b^3 \left (4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right )\right )+12 a (-b c+a d)^3 \log \left (a+b x^2\right )}{24 b^5} \]
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Time = 2.59 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.37
method | result | size |
norman | \(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 b^{4}}+\frac {d^{3} x^{8}}{8 b}-\frac {d^{2} \left (a d -3 b c \right ) x^{6}}{6 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{4}}{4 b^{3}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\) | \(157\) |
default | \(-\frac {-\frac {1}{4} d^{3} x^{8} b^{3}+\frac {1}{3} x^{6} a \,b^{2} d^{3}-x^{6} b^{3} c \,d^{2}-\frac {1}{2} x^{4} a^{2} b \,d^{3}+\frac {3}{2} x^{4} a \,b^{2} c \,d^{2}-\frac {3}{2} x^{4} b^{3} c^{2} d +a^{3} d^{3} x^{2}-3 a^{2} b c \,d^{2} x^{2}+3 a \,b^{2} c^{2} d \,x^{2}-b^{3} c^{3} x^{2}}{2 b^{4}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\) | \(177\) |
parallelrisch | \(\frac {3 b^{4} d^{3} x^{8}-4 x^{6} a \,b^{3} d^{3}+12 x^{6} b^{4} c \,d^{2}+6 x^{4} a^{2} b^{2} d^{3}-18 x^{4} a \,b^{3} c \,d^{2}+18 x^{4} b^{4} c^{2} d -12 a^{3} b \,d^{3} x^{2}+36 a^{2} b^{2} c \,d^{2} x^{2}-36 a \,b^{3} c^{2} d \,x^{2}+12 b^{4} c^{3} x^{2}+12 \ln \left (b \,x^{2}+a \right ) a^{4} d^{3}-36 \ln \left (b \,x^{2}+a \right ) a^{3} b c \,d^{2}+36 \ln \left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2} d -12 \ln \left (b \,x^{2}+a \right ) a \,b^{3} c^{3}}{24 b^{5}}\) | \(203\) |
risch | \(\frac {d^{3} x^{8}}{8 b}-\frac {x^{6} a \,d^{3}}{6 b^{2}}+\frac {x^{6} c \,d^{2}}{2 b}+\frac {x^{4} a^{2} d^{3}}{4 b^{3}}-\frac {3 x^{4} a c \,d^{2}}{4 b^{2}}+\frac {3 x^{4} c^{2} d}{4 b}-\frac {a^{3} d^{3} x^{2}}{2 b^{4}}+\frac {3 a^{2} c \,d^{2} x^{2}}{2 b^{3}}-\frac {3 a \,c^{2} d \,x^{2}}{2 b^{2}}+\frac {c^{3} x^{2}}{2 b}+\frac {a^{4} \ln \left (b \,x^{2}+a \right ) d^{3}}{2 b^{5}}-\frac {3 a^{3} \ln \left (b \,x^{2}+a \right ) c \,d^{2}}{2 b^{4}}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right ) c^{2} d}{2 b^{3}}-\frac {a \ln \left (b \,x^{2}+a \right ) c^{3}}{2 b^{2}}\) | \(205\) |
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Time = 0.23 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {3 \, b^{4} d^{3} x^{8} + 4 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{6} + 6 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{4} + 12 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2} - 12 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left (b x^{2} + a\right )}{24 \, b^{5}} \]
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Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.25 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {a \left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{5}} + x^{6} \left (- \frac {a d^{3}}{6 b^{2}} + \frac {c d^{2}}{2 b}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4 b^{3}} - \frac {3 a c d^{2}}{4 b^{2}} + \frac {3 c^{2} d}{4 b}\right ) + x^{2} \left (- \frac {a^{3} d^{3}}{2 b^{4}} + \frac {3 a^{2} c d^{2}}{2 b^{3}} - \frac {3 a c^{2} d}{2 b^{2}} + \frac {c^{3}}{2 b}\right ) + \frac {d^{3} x^{8}}{8 b} \]
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Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.46 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {3 \, b^{3} d^{3} x^{8} + 4 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{6} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + 12 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{24 \, b^{4}} - \frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.57 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {3 \, b^{3} d^{3} x^{8} + 12 \, b^{3} c d^{2} x^{6} - 4 \, a b^{2} d^{3} x^{6} + 18 \, b^{3} c^{2} d x^{4} - 18 \, a b^{2} c d^{2} x^{4} + 6 \, a^{2} b d^{3} x^{4} + 12 \, b^{3} c^{3} x^{2} - 36 \, a b^{2} c^{2} d x^{2} + 36 \, a^{2} b c d^{2} x^{2} - 12 \, a^{3} d^{3} x^{2}}{24 \, b^{4}} - \frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} \]
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Time = 5.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.55 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^2\,\left (\frac {c^3}{2\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{2\,b}\right )-x^6\,\left (\frac {a\,d^3}{6\,b^2}-\frac {c\,d^2}{2\,b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{4\,b}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3\right )}{2\,b^5}+\frac {d^3\,x^8}{8\,b} \]
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