Optimal. Leaf size=275 \[ \frac {7 e}{4 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x)}{12 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 b e^2 (a+b x)}{4 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 b^{3/2} e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 44, 53, 65,
214} \begin {gather*} \frac {35 b e^2 (a+b x)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}+\frac {35 e^2 (a+b x)}{12 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac {7 e}{4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac {35 b^{3/2} e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)^{5/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{5/2}} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e}{4 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{8 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e}{4 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x)}{12 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 b e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e}{4 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x)}{12 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 b e^2 (a+b x)}{4 (b d-a e)^4 \sqrt {d+e x} \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 ) \sqrt {d+e x}} \, dx}{8 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e}{4 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x)}{12 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 b e^2 (a+b x)}{4 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 b^2 e \left (a b+b^2 x\right )\right ) \text {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 )}{4 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 e}{4 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^2 (a+b x)}{12 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 b e^2 (a+b x)}{4 (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 b^{3/2} e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 188, normalized size = 0.68 \begin {gather*} \frac {e^2 (a+b x)^3 \left (\frac {-8 a^3 e^3+8 a^2 b e^2 (10 d+7 e x)+a b^2 e \left (39 d^2+238 d e x+175 e^2 x^2\right )+b^3 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )}{e^2 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {105 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}}\right )}{12 \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(387\) vs.
\(2(192)=384\).
time = 0.67, size = 388, normalized size = 1.41
method | result | size |
default | \(\frac {\left (105 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{4} e^{2} x^{2}+210 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{3} e^{2} x +105 \left (e x +d \right )^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} e^{2}+105 \sqrt {b \left (a e -b d \right )}\, b^{3} e^{3} x^{3}+175 \sqrt {b \left (a e -b d \right )}\, a \,b^{2} e^{3} x^{2}+140 \sqrt {b \left (a e -b d \right )}\, b^{3} d \,e^{2} x^{2}+56 \sqrt {b \left (a e -b d \right )}\, a^{2} b \,e^{3} x +238 \sqrt {b \left (a e -b d \right )}\, a \,b^{2} d \,e^{2} x +21 \sqrt {b \left (a e -b d \right )}\, b^{3} d^{2} e x -8 \sqrt {b \left (a e -b d \right )}\, a^{3} e^{3}+80 \sqrt {b \left (a e -b d \right )}\, a^{2} b d \,e^{2}+39 \sqrt {b \left (a e -b d \right )}\, a \,b^{2} d^{2} e -6 \sqrt {b \left (a e -b d \right )}\, b^{3} d^{3}\right ) \left (b x +a \right )}{12 \left (e x +d \right )^{\frac {3}{2}} \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(388\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 590 vs.
\(2 (201) = 402\).
time = 1.89, size = 1192, normalized size = 4.33 \begin {gather*} \left [\frac {105 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} e^{4} + 2 \, {\left (b^{3} d x^{3} + 2 \, a b^{2} d x^{2} + a^{2} b d x\right )} e^{3} + {\left (b^{3} d^{2} x^{2} + 2 \, a b^{2} d^{2} x + a^{2} b d^{2}\right )} e^{2}\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {2 \, b d - 2 \, {\left (b d - a e\right )} \sqrt {x e + d} \sqrt {\frac {b}{b d - a e}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (6 \, b^{3} d^{3} - {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} e^{3} - 2 \, {\left (70 \, b^{3} d x^{2} + 119 \, a b^{2} d x + 40 \, a^{2} b d\right )} e^{2} - 3 \, {\left (7 \, b^{3} d^{2} x + 13 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{6} d^{6} x^{2} + 2 \, a b^{5} d^{6} x + a^{2} b^{4} d^{6} + {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{4} + 3 \, a^{4} b^{2} d x^{3} - a^{6} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{4} + 4 \, a^{3} b^{3} d^{2} x^{3} - 9 \, a^{4} b^{2} d^{2} x^{2} - 6 \, a^{5} b d^{2} x + a^{6} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{4} - a^{2} b^{4} d^{3} x^{3} - 4 \, a^{3} b^{3} d^{3} x^{2} - a^{4} b^{2} d^{3} x + a^{5} b d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{4} - 6 \, a b^{5} d^{4} x^{3} - 9 \, a^{2} b^{4} d^{4} x^{2} + 4 \, a^{3} b^{3} d^{4} x + 6 \, a^{4} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x^{3} - 3 \, a^{2} b^{4} d^{5} x - 2 \, a^{3} b^{3} d^{5}\right )} e\right )}}, -\frac {105 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} e^{4} + 2 \, {\left (b^{3} d x^{3} + 2 \, a b^{2} d x^{2} + a^{2} b d x\right )} e^{3} + {\left (b^{3} d^{2} x^{2} + 2 \, a b^{2} d^{2} x + a^{2} b d^{2}\right )} e^{2}\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {x e + d} \sqrt {-\frac {b}{b d - a e}}}{b x e + b d}\right ) + {\left (6 \, b^{3} d^{3} - {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} e^{3} - 2 \, {\left (70 \, b^{3} d x^{2} + 119 \, a b^{2} d x + 40 \, a^{2} b d\right )} e^{2} - 3 \, {\left (7 \, b^{3} d^{2} x + 13 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{12 \, {\left (b^{6} d^{6} x^{2} + 2 \, a b^{5} d^{6} x + a^{2} b^{4} d^{6} + {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{4} + 3 \, a^{4} b^{2} d x^{3} - a^{6} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{4} + 4 \, a^{3} b^{3} d^{2} x^{3} - 9 \, a^{4} b^{2} d^{2} x^{2} - 6 \, a^{5} b d^{2} x + a^{6} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{4} - a^{2} b^{4} d^{3} x^{3} - 4 \, a^{3} b^{3} d^{3} x^{2} - a^{4} b^{2} d^{3} x + a^{5} b d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{4} - 6 \, a b^{5} d^{4} x^{3} - 9 \, a^{2} b^{4} d^{4} x^{2} + 4 \, a^{3} b^{3} d^{4} x + 6 \, a^{4} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x^{3} - 3 \, a^{2} b^{4} d^{5} x - 2 \, a^{3} b^{3} d^{5}\right )} e\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{\frac {5}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.89, size = 385, normalized size = 1.40 \begin {gather*} \frac {35 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (9 \, {\left (x e + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {11 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{2} - 13 \, \sqrt {x e + d} b^{3} d e^{2} + 13 \, \sqrt {x e + d} a b^{2} e^{3}}{4 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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