Integrand size = 19, antiderivative size = 162 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {(a e-c d x) (d+e x)^3}{a c \sqrt {a+c x^2}}-\frac {d e (d+e x)^2 \sqrt {a+c x^2}}{a c}-\frac {e \left (4 d \left (c d^2-4 a e^2\right )+e \left (2 c d^2-3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{2 a c^2}+\frac {3 e^2 \left (4 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}} \]
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Time = 0.10 (sec) , antiderivative size = 162, 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, 847, 794, 223, 212} \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (4 c d^2-a e^2\right )}{2 c^{5/2}}-\frac {e \sqrt {a+c x^2} \left (e x \left (2 c d^2-3 a e^2\right )+4 d \left (c d^2-4 a e^2\right )\right )}{2 a c^2}-\frac {(d+e x)^3 (a e-c d x)}{a c \sqrt {a+c x^2}}-\frac {d e \sqrt {a+c x^2} (d+e x)^2}{a c} \]
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Rule 212
Rule 223
Rule 753
Rule 794
Rule 847
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^3}{a c \sqrt {a+c x^2}}+\frac {\int \frac {(d+e x)^2 \left (3 a e^2-3 c d e x\right )}{\sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {(a e-c d x) (d+e x)^3}{a c \sqrt {a+c x^2}}-\frac {d e (d+e x)^2 \sqrt {a+c x^2}}{a c}+\frac {\int \frac {(d+e x) \left (15 a c d e^2-3 c e \left (2 c d^2-3 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{3 a c^2} \\ & = -\frac {(a e-c d x) (d+e x)^3}{a c \sqrt {a+c x^2}}-\frac {d e (d+e x)^2 \sqrt {a+c x^2}}{a c}-\frac {e \left (4 d \left (c d^2-4 a e^2\right )+e \left (2 c d^2-3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{2 a c^2}+\frac {\left (3 e^2 \left (4 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^3}{a c \sqrt {a+c x^2}}-\frac {d e (d+e x)^2 \sqrt {a+c x^2}}{a c}-\frac {e \left (4 d \left (c d^2-4 a e^2\right )+e \left (2 c d^2-3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{2 a c^2}+\frac {\left (3 e^2 \left (4 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c^2} \\ & = -\frac {(a e-c d x) (d+e x)^3}{a c \sqrt {a+c x^2}}-\frac {d e (d+e x)^2 \sqrt {a+c x^2}}{a c}-\frac {e \left (4 d \left (c d^2-4 a e^2\right )+e \left (2 c d^2-3 a e^2\right ) x\right ) \sqrt {a+c x^2}}{2 a c^2}+\frac {3 e^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {2 c^2 d^4 x+a^2 e^3 (16 d+3 e x)+a c e \left (-8 d^3-12 d^2 e x+8 d e^2 x^2+e^3 x^3\right )}{2 a c^2 \sqrt {a+c x^2}}-\frac {3 \left (4 c d^2 e^2-a e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{2 c^{5/2}} \]
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Time = 2.05 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {e^{3} \left (e x +8 d \right ) \sqrt {c \,x^{2}+a}}{2 c^{2}}-\frac {\frac {a \,e^{4} x}{\sqrt {c \,x^{2}+a}}-\frac {2 c^{2} d^{4} x}{a \sqrt {c \,x^{2}+a}}+\left (3 a c \,e^{4}-12 c^{2} d^{2} e^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )-\frac {8 a c d \,e^{3}-8 d^{3} e \,c^{2}}{c \sqrt {c \,x^{2}+a}}}{2 c^{2}}\) | \(156\) |
default | \(\frac {d^{4} x}{a \sqrt {c \,x^{2}+a}}+e^{4} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+4 d \,e^{3} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}\right )+6 d^{2} e^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )-\frac {4 d^{3} e}{c \sqrt {c \,x^{2}+a}}\) | \(183\) |
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Time = 0.28 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.27 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (4 \, a^{2} c d^{2} e^{2} - a^{3} e^{4} + {\left (4 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (a c^{2} e^{4} x^{3} + 8 \, a c^{2} d e^{3} x^{2} - 8 \, a c^{2} d^{3} e + 16 \, a^{2} c d e^{3} + {\left (2 \, c^{3} d^{4} - 12 \, a c^{2} d^{2} e^{2} + 3 \, a^{2} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}, -\frac {3 \, {\left (4 \, a^{2} c d^{2} e^{2} - a^{3} e^{4} + {\left (4 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (a c^{2} e^{4} x^{3} + 8 \, a c^{2} d e^{3} x^{2} - 8 \, a c^{2} d^{3} e + 16 \, a^{2} c d e^{3} + {\left (2 \, c^{3} d^{4} - 12 \, a c^{2} d^{2} e^{2} + 3 \, a^{2} c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}\right ] \]
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\[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {e^{4} x^{3}}{2 \, \sqrt {c x^{2} + a} c} + \frac {4 \, d e^{3} x^{2}}{\sqrt {c x^{2} + a} c} + \frac {d^{4} x}{\sqrt {c x^{2} + a} a} - \frac {6 \, d^{2} e^{2} x}{\sqrt {c x^{2} + a} c} + \frac {3 \, a e^{4} x}{2 \, \sqrt {c x^{2} + a} c^{2}} + \frac {6 \, d^{2} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}}} - \frac {3 \, a e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {5}{2}}} - \frac {4 \, d^{3} e}{\sqrt {c x^{2} + a} c} + \frac {8 \, a d e^{3}}{\sqrt {c x^{2} + a} c^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {e^{4} x}{c} + \frac {8 \, d e^{3}}{c}\right )} x + \frac {2 \, c^{4} d^{4} - 12 \, a c^{3} d^{2} e^{2} + 3 \, a^{2} c^{2} e^{4}}{a c^{4}}\right )} x - \frac {8 \, {\left (a c^{3} d^{3} e - 2 \, a^{2} c^{2} d e^{3}\right )}}{a c^{4}}}{2 \, \sqrt {c x^{2} + a}} - \frac {3 \, {\left (4 \, c d^{2} e^{2} - a e^{4}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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