Optimal. Leaf size=276 \[ -\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3}+\frac {3 c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2} d}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {3 c}{2 a^2 d \sqrt {a+c x^2}}+\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}-\frac {e^3 (a e+c d x)}{a d^3 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {1}{2 a d x^2 \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.24, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {961, 266, 51, 63, 208, 271, 191, 741, 12, 725, 206} \[ -\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {3 \sqrt {a+c x^2}}{2 a^2 d x^2}+\frac {3 c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2} d}-\frac {e^3 (a e+c d x)}{a d^3 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 191
Rule 206
Rule 208
Rule 266
Rule 271
Rule 725
Rule 741
Rule 961
Rubi steps
\begin {align*} \int \frac {1}{x^3 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\int \left (\frac {1}{d x^3 \left (a+c x^2\right )^{3/2}}-\frac {e}{d^2 x^2 \left (a+c x^2\right )^{3/2}}+\frac {e^2}{d^3 x \left (a+c x^2\right )^{3/2}}-\frac {e^3}{d^3 (d+e x) \left (a+c x^2\right )^{3/2}}\right ) \, dx\\ &=\frac {\int \frac {1}{x^3 \left (a+c x^2\right )^{3/2}} \, dx}{d}-\frac {e \int \frac {1}{x^2 \left (a+c x^2\right )^{3/2}} \, dx}{d^2}+\frac {e^2 \int \frac {1}{x \left (a+c x^2\right )^{3/2}} \, dx}{d^3}-\frac {e^3 \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx}{d^3}\\ &=\frac {e}{a d^2 x \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d}+\frac {(2 c e) \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{a d^2}+\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{x (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d^3}-\frac {e^3 \int \frac {a e^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a d^3 \left (c d^2+a e^2\right )}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d}+\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d^3}-\frac {e^5 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^3 \left (c d^2+a e^2\right )}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {3 \sqrt {a+c x^2}}{2 a^2 d x^2}-\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a^2 d}+\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a c d^3}+\frac {e^5 \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 \left (c d^2+a e^2\right )}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {3 \sqrt {a+c x^2}}{2 a^2 d x^2}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \left (c d^2+a e^2\right )^{3/2}}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 a^2 d}\\ &=\frac {e^2}{a d^3 \sqrt {a+c x^2}}+\frac {1}{a d x^2 \sqrt {a+c x^2}}+\frac {e}{a d^2 x \sqrt {a+c x^2}}+\frac {2 c e x}{a^2 d^2 \sqrt {a+c x^2}}-\frac {e^3 (a e+c d x)}{a d^3 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {3 \sqrt {a+c x^2}}{2 a^2 d x^2}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \left (c d^2+a e^2\right )^{3/2}}+\frac {3 c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2} d}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^3}\\ \end {align*}
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Mathematica [C] time = 0.34, size = 203, normalized size = 0.74 \[ \frac {-\frac {c d^2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {c x^2}{a}+1\right )}{a^2 \sqrt {a+c x^2}}+\frac {d e \left (a+2 c x^2\right )}{a^2 x \sqrt {a+c x^2}}+\frac {e^5 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {e^3 (a e+c d x)}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {e^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c x^2}{a}+1\right )}{a \sqrt {a+c x^2}}}{d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.13, size = 1943, normalized size = 7.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 358, normalized size = 1.30 \[ \frac {\frac {{\left (a^{2} c^{3} d^{2} e + a^{3} c^{2} e^{3}\right )} x}{a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4}} - \frac {a^{2} c^{3} d^{3} + a^{3} c^{2} d e^{2}}{a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{5}}{{\left (c d^{5} + a d^{3} e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {{\left (3 \, c d^{2} - 2 \, a e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} d^{3}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c d - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a \sqrt {c} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c d + 2 \, a^{2} \sqrt {c} e}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 439, normalized size = 1.59 \[ -\frac {c \,e^{3} x}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, a \,d^{2}}+\frac {e^{4} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{3}}-\frac {e^{4}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{3}}+\frac {2 c e x}{\sqrt {c \,x^{2}+a}\, a^{2} d^{2}}-\frac {e^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {3}{2}} d^{3}}+\frac {3 c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}} d}+\frac {e^{2}}{\sqrt {c \,x^{2}+a}\, a \,d^{3}}-\frac {3 c}{2 \sqrt {c \,x^{2}+a}\, a^{2} d}+\frac {e}{\sqrt {c \,x^{2}+a}\, a \,d^{2} x}-\frac {1}{2 \sqrt {c \,x^{2}+a}\, a d \,x^{2}} \]
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 \[ \int \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^3\,{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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