Optimal. Leaf size=160 \[ -\frac {\sqrt {a} e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {e \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3}-\frac {\sqrt {a+c x^2}}{2 d x^2}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d} \]
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Rubi [A] time = 0.21, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {961, 266, 47, 63, 208, 277, 217, 206, 50, 735, 844, 725} \[ -\frac {\sqrt {a} e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}+\frac {e \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3}+\frac {e \sqrt {a+c x^2}}{d^2 x}-\frac {\sqrt {a+c x^2}}{2 d x^2}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 277
Rule 725
Rule 735
Rule 844
Rule 961
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{x^3 (d+e x)} \, dx &=\int \left (\frac {\sqrt {a+c x^2}}{d x^3}-\frac {e \sqrt {a+c x^2}}{d^2 x^2}+\frac {e^2 \sqrt {a+c x^2}}{d^3 x}-\frac {e^3 \sqrt {a+c x^2}}{d^3 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {a+c x^2}}{x^3} \, dx}{d}-\frac {e \int \frac {\sqrt {a+c x^2}}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\sqrt {a+c x^2}}{x} \, dx}{d^3}-\frac {e^3 \int \frac {\sqrt {a+c x^2}}{d+e x} \, dx}{d^3}\\ &=-\frac {e^2 \sqrt {a+c x^2}}{d^3}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{2 d}-\frac {(c e) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^2}+\frac {e^2 \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^3}-\frac {e^2 \int \frac {a e-c d x}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^3}\\ &=-\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 d}+\frac {(c e) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^2}-\frac {(c e) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^2}+\frac {\left (a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (e \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^3}\\ &=-\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}-\frac {\sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{d^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 d}+\frac {(c e) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^2}+\frac {\left (a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^3}+\frac {\left (e \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{d^3}\\ &=-\frac {\sqrt {a+c x^2}}{2 d x^2}+\frac {e \sqrt {a+c x^2}}{d^2 x}+\frac {e \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d}-\frac {\sqrt {a} e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 283, normalized size = 1.77 \[ -\frac {-2 e x^2 \sqrt {a+c x^2} \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )+c d^2 x^2 \sqrt {\frac {c x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )+2 \sqrt {a} \sqrt {c} d e x^2 \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )-2 \sqrt {c} d e x^2 \sqrt {a+c x^2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+2 \sqrt {a} e^2 x^2 \sqrt {a+c x^2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+a d^2-2 a d e x+c d^2 x^2-2 c d e x^3}{2 d^3 x^2 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 726, normalized size = 4.54 \[ \left [\frac {2 \, \sqrt {c d^{2} + a e^{2}} a e x^{2} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + {\left (c d^{2} + 2 \, a e^{2}\right )} \sqrt {a} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, a d e x - a d^{2}\right )} \sqrt {c x^{2} + a}}{4 \, a d^{3} x^{2}}, \frac {4 \, \sqrt {-c d^{2} - a e^{2}} a e x^{2} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (c d^{2} + 2 \, a e^{2}\right )} \sqrt {a} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, a d e x - a d^{2}\right )} \sqrt {c x^{2} + a}}{4 \, a d^{3} x^{2}}, \frac {\sqrt {c d^{2} + a e^{2}} a e x^{2} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + {\left (c d^{2} + 2 \, a e^{2}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (2 \, a d e x - a d^{2}\right )} \sqrt {c x^{2} + a}}{2 \, a d^{3} x^{2}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} a e x^{2} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (c d^{2} + 2 \, a e^{2}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (2 \, a d e x - a d^{2}\right )} \sqrt {c x^{2} + a}}{2 \, a d^{3} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 230, normalized size = 1.44 \[ -\frac {2 \, {\left (c d^{2} e + a e^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} d^{3}} + \frac {{\left (c d^{2} + 2 \, a e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} d^{3}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c d - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a \sqrt {c} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c d + 2 \, a^{2} \sqrt {c} e}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 567, normalized size = 3.54 \[ \frac {a \,e^{2} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{3}}+\frac {c \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d}-\frac {\sqrt {a}\, e^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{d^{3}}-\frac {c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 \sqrt {a}\, d}-\frac {\sqrt {c}\, e \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{d^{2}}+\frac {\sqrt {c}\, e \ln \left (\frac {-\frac {c d}{e}+\left (x +\frac {d}{e}\right ) c}{\sqrt {c}}+\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{d^{2}}-\frac {\sqrt {c \,x^{2}+a}\, c e x}{a \,d^{2}}+\frac {\sqrt {c \,x^{2}+a}\, c}{2 a d}+\frac {\sqrt {c \,x^{2}+a}\, e^{2}}{d^{3}}-\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{2}}{d^{3}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} e}{a \,d^{2} x}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{2 a d \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c\,x^2+a}}{x^3\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + c x^{2}}}{x^{3} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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