Optimal. Leaf size=173 \[ -\frac {35 e^3}{8 \sqrt {d+e x} (b d-a e)^4}+\frac {35 \sqrt {b} e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{9/2}}-\frac {35 e^2}{24 (a+b x) \sqrt {d+e x} (b d-a e)^3}+\frac {7 e}{12 (a+b x)^2 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)} \]
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Rubi [A] time = 0.08, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac {35 e^3}{8 \sqrt {d+e x} (b d-a e)^4}-\frac {35 e^2}{24 (a+b x) \sqrt {d+e x} (b d-a e)^3}+\frac {35 \sqrt {b} e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{9/2}}+\frac {7 e}{12 (a+b x)^2 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^4 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{3 (b d-a e) (a+b x)^3 \sqrt {d+e x}}-\frac {(7 e) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{6 (b d-a e)}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^3 \sqrt {d+e x}}+\frac {7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}+\frac {\left (35 e^2\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{24 (b d-a e)^2}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^3 \sqrt {d+e x}}+\frac {7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}-\frac {35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt {d+e x}}-\frac {\left (35 e^3\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^3}\\ &=-\frac {35 e^3}{8 (b d-a e)^4 \sqrt {d+e x}}-\frac {1}{3 (b d-a e) (a+b x)^3 \sqrt {d+e x}}+\frac {7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}-\frac {35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt {d+e x}}-\frac {\left (35 b e^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^4}\\ &=-\frac {35 e^3}{8 (b d-a e)^4 \sqrt {d+e x}}-\frac {1}{3 (b d-a e) (a+b x)^3 \sqrt {d+e x}}+\frac {7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}-\frac {35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt {d+e x}}-\frac {\left (35 b e^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 (b d-a e)^4}\\ &=-\frac {35 e^3}{8 (b d-a e)^4 \sqrt {d+e x}}-\frac {1}{3 (b d-a e) (a+b x)^3 \sqrt {d+e x}}+\frac {7 e}{12 (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}-\frac {35 e^2}{24 (b d-a e)^3 (a+b x) \sqrt {d+e x}}+\frac {35 \sqrt {b} e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.29 \[ -\frac {2 e^3 \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{\sqrt {d+e x} (a e-b d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.30, size = 1204, normalized size = 6.96 \[ \left [\frac {105 \, {\left (b^{3} e^{4} x^{4} + a^{3} d e^{3} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (105 \, b^{3} e^{3} x^{3} + 8 \, b^{3} d^{3} - 38 \, a b^{2} d^{2} e + 87 \, a^{2} b d e^{2} + 48 \, a^{3} e^{3} + 35 \, {\left (b^{3} d e^{2} + 8 \, a b^{2} e^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} d^{2} e - 14 \, a b^{2} d e^{2} - 33 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}}, \frac {105 \, {\left (b^{3} e^{4} x^{4} + a^{3} d e^{3} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 3 \, {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{2} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (105 \, b^{3} e^{3} x^{3} + 8 \, b^{3} d^{3} - 38 \, a b^{2} d^{2} e + 87 \, a^{2} b d e^{2} + 48 \, a^{3} e^{3} + 35 \, {\left (b^{3} d e^{2} + 8 \, a b^{2} e^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} d^{2} e - 14 \, a b^{2} d e^{2} - 33 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{4} d^{5} - 4 \, a^{4} b^{3} d^{4} e + 6 \, a^{5} b^{2} d^{3} e^{2} - 4 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4} + {\left (b^{7} d^{4} e - 4 \, a b^{6} d^{3} e^{2} + 6 \, a^{2} b^{5} d^{2} e^{3} - 4 \, a^{3} b^{4} d e^{4} + a^{4} b^{3} e^{5}\right )} x^{4} + {\left (b^{7} d^{5} - a b^{6} d^{4} e - 6 \, a^{2} b^{5} d^{3} e^{2} + 14 \, a^{3} b^{4} d^{2} e^{3} - 11 \, a^{4} b^{3} d e^{4} + 3 \, a^{5} b^{2} e^{5}\right )} x^{3} + 3 \, {\left (a b^{6} d^{5} - 3 \, a^{2} b^{5} d^{4} e + 2 \, a^{3} b^{4} d^{3} e^{2} + 2 \, a^{4} b^{3} d^{2} e^{3} - 3 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} d^{5} - 11 \, a^{3} b^{4} d^{4} e + 14 \, a^{4} b^{3} d^{3} e^{2} - 6 \, a^{5} b^{2} d^{2} e^{3} - a^{6} b d e^{4} + a^{7} e^{5}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 324, normalized size = 1.87 \[ -\frac {35 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, e^{3}}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {x e + d}} - \frac {57 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{3} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{3} + 87 \, \sqrt {x e + d} b^{3} d^{2} e^{3} + 136 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} e^{4} - 174 \, \sqrt {x e + d} a b^{2} d e^{4} + 87 \, \sqrt {x e + d} a^{2} b e^{5}}{24 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 292, normalized size = 1.69 \[ -\frac {29 \sqrt {e x +d}\, a^{2} b \,e^{5}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {29 \sqrt {e x +d}\, a \,b^{2} d \,e^{4}}{4 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {29 \sqrt {e x +d}\, b^{3} d^{2} e^{3}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {17 \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} e^{4}}{3 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} b^{3} d \,e^{3}}{3 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {19 \left (e x +d \right )^{\frac {5}{2}} b^{3} e^{3}}{8 \left (a e -b d \right )^{4} \left (b e x +a e \right )^{3}}-\frac {35 b \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right )^{4} \sqrt {\left (a e -b d \right ) b}}-\frac {2 e^{3}}{\left (a e -b d \right )^{4} \sqrt {e x +d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 294, normalized size = 1.70 \[ -\frac {\frac {2\,e^3}{a\,e-b\,d}+\frac {35\,b^2\,e^3\,{\left (d+e\,x\right )}^2}{3\,{\left (a\,e-b\,d\right )}^3}+\frac {35\,b^3\,e^3\,{\left (d+e\,x\right )}^3}{8\,{\left (a\,e-b\,d\right )}^4}+\frac {77\,b\,e^3\,\left (d+e\,x\right )}{8\,{\left (a\,e-b\,d\right )}^2}}{\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )+b^3\,{\left (d+e\,x\right )}^{7/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}-\frac {35\,\sqrt {b}\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}}\right )}{8\,{\left (a\,e-b\,d\right )}^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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