Optimal. Leaf size=206 \[ \frac {315 e^4}{64 \sqrt {d+e x} (b d-a e)^5}-\frac {315 \sqrt {b} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2}}+\frac {105 e^3}{64 (a+b x) \sqrt {d+e x} (b d-a e)^4}-\frac {21 e^2}{32 (a+b x)^2 \sqrt {d+e x} (b d-a e)^3}+\frac {3 e}{8 (a+b x)^3 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)} \]
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Rubi [A] time = 0.12, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \[ \frac {315 e^4}{64 \sqrt {d+e x} (b d-a e)^5}+\frac {105 e^3}{64 (a+b x) \sqrt {d+e x} (b d-a e)^4}-\frac {21 e^2}{32 (a+b x)^2 \sqrt {d+e x} (b d-a e)^3}-\frac {315 \sqrt {b} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2}}+\frac {3 e}{8 (a+b x)^3 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{4 (a+b x)^4 \sqrt {d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}-\frac {(9 e) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{8 (b d-a e)}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}+\frac {\left (21 e^2\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{16 (b d-a e)^2}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}-\frac {\left (105 e^3\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{64 (b d-a e)^3}\\ &=-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}+\frac {\left (315 e^4\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4}\\ &=\frac {315 e^4}{64 (b d-a e)^5 \sqrt {d+e x}}-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}+\frac {\left (315 b e^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{128 (b d-a e)^5}\\ &=\frac {315 e^4}{64 (b d-a e)^5 \sqrt {d+e x}}-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}+\frac {\left (315 b e^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 (b d-a e)^5}\\ &=\frac {315 e^4}{64 (b d-a e)^5 \sqrt {d+e x}}-\frac {1}{4 (b d-a e) (a+b x)^4 \sqrt {d+e x}}+\frac {3 e}{8 (b d-a e)^2 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2}{32 (b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}+\frac {105 e^3}{64 (b d-a e)^4 (a+b x) \sqrt {d+e x}}-\frac {315 \sqrt {b} e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.24 \[ -\frac {2 e^4 \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{\sqrt {d+e x} (a e-b d)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 1734, normalized size = 8.42 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 440, normalized size = 2.14 \[ \frac {315 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, e^{4}}{{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {x e + d}} + \frac {187 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} e^{4} - 643 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} d e^{4} + 765 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} d^{2} e^{4} - 325 \, \sqrt {x e + d} b^{4} d^{3} e^{4} + 643 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{3} e^{5} - 1530 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{3} d e^{5} + 975 \, \sqrt {x e + d} a b^{3} d^{2} e^{5} + 765 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{2} e^{6} - 975 \, \sqrt {x e + d} a^{2} b^{2} d e^{6} + 325 \, \sqrt {x e + d} a^{3} b e^{7}}{64 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 446, normalized size = 2.17 \[ -\frac {325 \sqrt {e x +d}\, a^{3} b \,e^{7}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {975 \sqrt {e x +d}\, a^{2} b^{2} d \,e^{6}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {975 \sqrt {e x +d}\, a \,b^{3} d^{2} e^{5}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {325 \sqrt {e x +d}\, b^{4} d^{3} e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {765 \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{2} e^{6}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {765 \left (e x +d \right )^{\frac {3}{2}} a \,b^{3} d \,e^{5}}{32 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {765 \left (e x +d \right )^{\frac {3}{2}} b^{4} d^{2} e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {643 \left (e x +d \right )^{\frac {5}{2}} a \,b^{3} e^{5}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}+\frac {643 \left (e x +d \right )^{\frac {5}{2}} b^{4} d \,e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {187 \left (e x +d \right )^{\frac {7}{2}} b^{4} e^{4}}{64 \left (a e -b d \right )^{5} \left (b e x +a e \right )^{4}}-\frac {315 b \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \left (a e -b d \right )^{5} \sqrt {\left (a e -b d \right ) b}}-\frac {2 e^{4}}{\left (a e -b d \right )^{5} \sqrt {e x +d}} \]
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 = 2.39, size = 398, normalized size = 1.93 \[ -\frac {\frac {2\,e^4}{a\,e-b\,d}+\frac {1533\,b^2\,e^4\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}+\frac {1155\,b^3\,e^4\,{\left (d+e\,x\right )}^3}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {315\,b^4\,e^4\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {837\,b\,e^4\,\left (d+e\,x\right )}{64\,{\left (a\,e-b\,d\right )}^2}}{b^4\,{\left (d+e\,x\right )}^{9/2}-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{7/2}+\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )+{\left (d+e\,x\right )}^{5/2}\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )}-\frac {315\,\sqrt {b}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^{11/2}}\right )}{64\,{\left (a\,e-b\,d\right )}^{11/2}} \]
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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