Optimal. Leaf size=278 \[ -\frac {35 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac {35 e^3 \sqrt {d+e x}}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {35 e^2 \sqrt {d+e x}}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {7 e \sqrt {d+e x}}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {\sqrt {d+e x}}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.14, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {646, 51, 63, 208} \[ \frac {35 e^3 \sqrt {d+e x}}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {35 e^2 \sqrt {d+e x}}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {35 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac {7 e \sqrt {d+e x}}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {\sqrt {d+e x}}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^5 \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b^3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 \sqrt {d+e x}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^3 \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^4 (a+b x) \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} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 65, normalized size = 0.23 \[ -\frac {2 e^4 (a+b x) \sqrt {d+e x} \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{\sqrt {(a+b x)^2} (b d-a e)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 1325, normalized size = 4.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 551, normalized size = 1.98 \[ \frac {35 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{4} d^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {105 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 385 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 511 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt {x e + d} b^{3} d^{3} e^{4} + 385 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 1022 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} + 837 \, \sqrt {x e + d} a b^{2} d^{2} e^{5} + 511 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{6} - 837 \, \sqrt {x e + d} a^{2} b d e^{6} + 279 \, \sqrt {x e + d} a^{3} e^{7}}{192 \, {\left (b^{4} d^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 497, normalized size = 1.79 \[ \frac {\left (105 b^{4} e^{4} x^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+420 a \,b^{3} e^{4} x^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+630 a^{2} b^{2} e^{4} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+420 a^{3} b \,e^{4} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 a^{4} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+105 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} e^{3} x^{3}+385 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e^{3} x^{2}-70 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d \,e^{2} x^{2}+511 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b \,e^{3} x -252 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d \,e^{2} x +56 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{2} e x +279 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} e^{3}-326 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} b d \,e^{2}+200 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} d^{2} e -48 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{3} d^{3}\right ) \left (b x +a \right )}{192 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} \sqrt {e x + d}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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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