Integrand size = 19, antiderivative size = 191 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {5 d e^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \]
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Time = 0.11 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {753, 833, 794, 223, 212} \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \sqrt {a+c x^2} \left (4 \left (-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )+c d e x \left (7 a e^2+2 c d^2\right )\right )}{3 a^2 c^3}+\frac {5 d e^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
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Rule 212
Rule 223
Rule 753
Rule 794
Rule 833
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(d+e x)^3 \left (2 \left (c d^2+2 a e^2\right )-2 c d e x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {(d+e x) \left (-2 a e^2 \left (c d^2-4 a e^2\right )-2 c d e \left (2 c d^2+7 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {\left (5 d e^4\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c^2} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {\left (5 d e^4\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c^2} \\ & = -\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {5 d e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {8 a^4 e^5+2 c^4 d^5 x^3+a c^3 d^3 x \left (3 d^2+10 e^2 x^2\right )+a^3 c e^3 \left (-20 d^2-15 d e x+12 e^2 x^2\right )+a^2 c^2 e \left (-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right )-15 a^2 \sqrt {c} d e^4 \left (a+c x^2\right )^{3/2} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{3 a^2 c^3 \left (a+c x^2\right )^{3/2}} \]
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Time = 2.36 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.47
method | result | size |
default | \(d^{5} \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )+e^{5} \left (\frac {x^{4}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{c}\right )+5 d \,e^{4} \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}}{c}\right )-\frac {5 d^{4} e}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+10 d^{2} e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )+10 d^{3} e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )\) | \(281\) |
risch | \(\frac {e^{5} \sqrt {c \,x^{2}+a}}{c^{3}}+\frac {\frac {5 d \,e^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}+\frac {\left (-15 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -4 \sqrt {-a c}\, a^{2} e^{5}+20 \sqrt {-a c}\, a c \,d^{2} e^{3}+d^{5} c^{3}\right ) \sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{4 a^{2} c^{2} \left (x -\frac {\sqrt {-a c}}{c}\right )}-\frac {\left (-4 \sqrt {-a c}\, a^{2} e^{5}+20 \sqrt {-a c}\, a c \,d^{2} e^{3}+15 d \,e^{4} a^{2} c -10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{4 a^{2} c^{2} \left (x +\frac {\sqrt {-a c}}{c}\right )}+\frac {\left (-\sqrt {-a c}\, a^{2} e^{5}+10 \sqrt {-a c}\, a c \,d^{2} e^{3}-5 \sqrt {-a c}\, c^{2} d^{4} e -5 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \left (-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x -\frac {\sqrt {-a c}}{c}\right )}\right )}{4 a \,c^{2}}+\frac {\left (\sqrt {-a c}\, a^{2} e^{5}-10 \sqrt {-a c}\, a c \,d^{2} e^{3}+5 \sqrt {-a c}\, c^{2} d^{4} e -5 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \left (\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x +\frac {\sqrt {-a c}}{c}\right )}\right )}{4 a \,c^{2}}}{c^{2}}\) | \(707\) |
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Time = 0.29 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.55 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{2} c^{2} d e^{4} x^{4} + 2 \, a^{3} c d e^{4} x^{2} + a^{4} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \, {\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}, -\frac {15 \, {\left (a^{2} c^{2} d e^{4} x^{4} + 2 \, a^{3} c d e^{4} x^{2} + a^{4} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \, {\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}\right ] \]
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\[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{5}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {5}{3} \, d e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, a}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}}\right )} + \frac {e^{5} x^{4}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {10 \, d^{2} e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {4 \, a e^{5} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, d^{5} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{5} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {10 \, d^{3} e^{2} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {10 \, d^{3} e^{2} x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {5 \, d e^{4} x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {5 \, d e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} - \frac {5 \, d^{4} e}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {20 \, a d^{2} e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} + \frac {8 \, a^{2} e^{5}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {5 \, d e^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {5}{2}}} + \frac {{\left ({\left ({\left (\frac {3 \, e^{5} x}{c} + \frac {2 \, {\left (c^{6} d^{5} + 5 \, a c^{5} d^{3} e^{2} - 10 \, a^{2} c^{4} d e^{4}\right )}}{a^{2} c^{5}}\right )} x - \frac {6 \, {\left (5 \, a^{2} c^{4} d^{2} e^{3} - 2 \, a^{3} c^{3} e^{5}\right )}}{a^{2} c^{5}}\right )} x + \frac {3 \, {\left (a c^{5} d^{5} - 5 \, a^{3} c^{3} d e^{4}\right )}}{a^{2} c^{5}}\right )} x - \frac {5 \, a^{2} c^{4} d^{4} e + 20 \, a^{3} c^{3} d^{2} e^{3} - 8 \, a^{4} c^{2} e^{5}}{a^{2} c^{5}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^5}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]
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