Integrand size = 19, antiderivative size = 161 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {d e \left (2 c d^2+5 a e^2\right ) \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {e^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 161, 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, 655, 223, 212} \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {d e \sqrt {a+c x^2} \left (5 a e^2+2 c d^2\right )}{3 a^2 c^2}-\frac {(d+e x) \left (a e \left (3 a e^2+c d^2\right )-2 c d x \left (2 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt {a+c x^2}}+\frac {e^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}-\frac {(d+e x)^3 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
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Rule 212
Rule 223
Rule 655
Rule 753
Rule 833
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(d+e x)^2 \left (2 c d^2+3 a e^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)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {3 a^2 e^4-c d e \left (2 c d^2+5 a e^2\right ) x}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {d e \left (2 c d^2+5 a e^2\right ) \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {e^4 \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c^2} \\ & = -\frac {(a e-c d x) (d+e x)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {d e \left (2 c d^2+5 a e^2\right ) \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {e^4 \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)^3}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {(d+e x) \left (a e \left (c d^2+3 a e^2\right )-2 c d \left (c d^2+2 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {d e \left (2 c d^2+5 a e^2\right ) \sqrt {a+c x^2}}{3 a^2 c^2}+\frac {e^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {2 c^3 d^4 x^3-a^3 e^3 (8 d+3 e x)+3 a c^2 d^2 x \left (d^2+2 e^2 x^2\right )-4 a^2 c e \left (d^3+3 d e^2 x^2+e^3 x^3\right )}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}}-\frac {e^4 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{5/2}} \]
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Time = 2.20 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.35
method | result | size |
default | \(d^{4} \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )+e^{4} \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}}{c}\right )+4 d \,e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )+6 d^{2} e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )-\frac {4 d^{3} e}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(217\) |
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Time = 0.30 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (12 \, a^{2} c^{2} d e^{3} x^{2} + 4 \, a^{2} c^{2} d^{3} e + 8 \, a^{3} c d e^{3} - 2 \, {\left (c^{4} d^{4} + 3 \, a c^{3} d^{2} e^{2} - 2 \, a^{2} c^{2} e^{4}\right )} x^{3} - 3 \, {\left (a c^{3} d^{4} - a^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}, -\frac {3 \, {\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (12 \, a^{2} c^{2} d e^{3} x^{2} + 4 \, a^{2} c^{2} d^{3} e + 8 \, a^{3} c d e^{3} - 2 \, {\left (c^{4} d^{4} + 3 \, a c^{3} d^{2} e^{2} - 2 \, a^{2} c^{2} e^{4}\right )} x^{3} - 3 \, {\left (a c^{3} d^{4} - a^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}\right ] \]
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\[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, a}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}}\right )} - \frac {4 \, d e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, d^{4} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{4} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {2 \, d^{2} e^{2} x}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, d^{2} e^{2} x}{\sqrt {c x^{2} + a} a c} - \frac {e^{4} x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} - \frac {4 \, d^{3} e}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {8 \, a d e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {e^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {5}{2}}} - \frac {{\left (2 \, {\left (\frac {6 \, d e^{3}}{c} - \frac {{\left (c^{5} d^{4} + 3 \, a c^{4} d^{2} e^{2} - 2 \, a^{2} c^{3} e^{4}\right )} x}{a^{2} c^{4}}\right )} x - \frac {3 \, {\left (a c^{4} d^{4} - a^{3} c^{2} e^{4}\right )}}{a^{2} c^{4}}\right )} x + \frac {4 \, {\left (a^{2} c^{3} d^{3} e + 2 \, a^{3} c^{2} d e^{3}\right )}}{a^{2} c^{4}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]
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