Optimal. Leaf size=180 \[ -\frac {35 e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2}}+\frac {35 e^3 \sqrt {d+e x}}{64 (a+b x) (b d-a e)^4}-\frac {35 e^2 \sqrt {d+e x}}{96 (a+b x)^2 (b d-a e)^3}+\frac {7 e \sqrt {d+e x}}{24 (a+b x)^3 (b d-a e)^2}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)} \]
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Rubi [A] time = 0.09, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \begin {gather*} \frac {35 e^3 \sqrt {d+e x}}{64 (a+b x) (b d-a e)^4}-\frac {35 e^2 \sqrt {d+e x}}{96 (a+b x)^2 (b d-a e)^3}-\frac {35 e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2}}+\frac {7 e \sqrt {d+e x}}{24 (a+b x)^3 (b d-a e)^2}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)} \end {gather*}
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}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 \sqrt {d+e x}} \, dx\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^4}-\frac {(7 e) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{8 (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^4}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^3}+\frac {\left (35 e^2\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{48 (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^4}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x)^2}-\frac {\left (35 e^3\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{64 (b d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^4}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x)^2}+\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)}+\frac {\left (35 e^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{128 (b d-a e)^4}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^4}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x)^2}+\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)}+\frac {\left (35 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)^4}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^4}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x)^2}+\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)}-\frac {35 e^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.28 \begin {gather*} \frac {2 e^4 \sqrt {d+e x} \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};-\frac {b (d+e x)}{a e-b d}\right )}{(a e-b d)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.52, size = 223, normalized size = 1.24 \begin {gather*} \frac {e^4 \sqrt {d+e x} \left (279 a^3 e^3+511 a^2 b e^2 (d+e x)-837 a^2 b d e^2+837 a b^2 d^2 e+385 a b^2 e (d+e x)^2-1022 a b^2 d e (d+e x)-279 b^3 d^3+511 b^3 d^2 (d+e x)+105 b^3 (d+e x)^3-385 b^3 d (d+e x)^2\right )}{192 (b d-a e)^4 (-a e-b (d+e x)+b d)^4}-\frac {35 e^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 \sqrt {b} (a e-b d)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 1325, normalized size = 7.36
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 331, normalized size = 1.84 \begin {gather*} \frac {35 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\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 {105 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 385 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 511 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt {x e + d} b^{3} d^{3} e^{4} + 385 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 1022 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} + 837 \, \sqrt {x e + d} a b^{2} d^{2} e^{5} + 511 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{6} - 837 \, \sqrt {x e + d} a^{2} b d e^{6} + 279 \, \sqrt {x e + d} a^{3} e^{7}}{192 \, {\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 )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 179, normalized size = 0.99 \begin {gather*} \frac {35 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 )^{4} \sqrt {\left (a e -b d \right ) b}}+\frac {\sqrt {e x +d}\, e^{4}}{4 \left (a e -b d \right ) \left (b e x +a e \right )^{4}}+\frac {7 \sqrt {e x +d}\, e^{4}}{24 \left (a e -b d \right )^{2} \left (b e x +a e \right )^{3}}+\frac {35 \sqrt {e x +d}\, e^{4}}{96 \left (a e -b d \right )^{3} \left (b e x +a e \right )^{2}}+\frac {35 \sqrt {e x +d}\, e^{4}}{64 \left (a e -b d \right )^{4} \left (b e x +a e \right )} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.15, size = 307, normalized size = 1.71 \begin {gather*} \frac {\frac {93\,e^4\,\sqrt {d+e\,x}}{64\,\left (a\,e-b\,d\right )}+\frac {385\,b^2\,e^4\,{\left (d+e\,x\right )}^{5/2}}{192\,{\left (a\,e-b\,d\right )}^3}+\frac {35\,b^3\,e^4\,{\left (d+e\,x\right )}^{7/2}}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {511\,b\,e^4\,{\left (d+e\,x\right )}^{3/2}}{192\,{\left (a\,e-b\,d\right )}^2}}{b^4\,{\left (d+e\,x\right )}^4-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^3-\left (d+e\,x\right )\,\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 )+a^4\,e^4+b^4\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e-4\,a^3\,b\,d\,e^3}+\frac {35\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{64\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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