Optimal. Leaf size=270 \[ \frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 270, 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, 43, 44, 65,
214} \begin {gather*} \frac {3 e^3 \sqrt {d+e x}}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\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 {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (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}{16 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (3 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 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 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^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e^3 \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 )}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 e^3 \sqrt {d+e x}}{64 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2 \sqrt {d+e x}}{32 b^2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.90, size = 194, normalized size = 0.72 \begin {gather*} \frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3 a^3 e^3+a^2 b e^2 (2 d+11 e x)-a b^2 e \left (24 d^2+44 d e x+11 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+2 d e^2 x^2-3 e^3 x^3\right )\right )}{e^4 (b d-a e)^2 (a+b x)^4}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}}\right )}{64 b^{5/2} \left ((a+b x)^2\right )^{5/2}} \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. \(476\) vs.
\(2(187)=374\).
time = 0.68, size = 477, normalized size = 1.77
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{4} e^{4} x^{4}+12 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{3} e^{4} x^{3}+3 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}+18 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} e^{4} x^{2}+11 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} a \,b^{2} e -11 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} b^{3} d +12 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b \,e^{4} x -11 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}+22 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e -11 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} e^{4}-3 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{3} e^{3}+9 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b d \,e^{2}-9 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} d^{2} e +3 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d^{3}\right )}{64 \sqrt {b \left (a e -b d \right )}\, b^{2} \left (a e -b d \right )^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(477\) |
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. 492 vs.
\(2 (194) = 388\).
time = 2.75, size = 998, normalized size = 3.70 \begin {gather*} \left [\frac {3 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {b^{2} d - a b e} e^{4} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) - 2 \, {\left (16 \, b^{5} d^{4} + {\left (3 \, a b^{4} x^{3} + 11 \, a^{2} b^{3} x^{2} - 11 \, a^{3} b^{2} x - 3 \, a^{4} b\right )} e^{4} - {\left (3 \, b^{5} d x^{3} + 13 \, a b^{4} d x^{2} - 55 \, a^{2} b^{3} d x - a^{3} b^{2} d\right )} e^{3} + 2 \, {\left (b^{5} d^{2} x^{2} - 34 \, a b^{4} d^{2} x + 13 \, a^{2} b^{3} d^{2}\right )} e^{2} + 8 \, {\left (3 \, b^{5} d^{3} x - 5 \, a b^{4} d^{3}\right )} e\right )} \sqrt {x e + d}}{128 \, {\left (b^{10} d^{3} x^{4} + 4 \, a b^{9} d^{3} x^{3} + 6 \, a^{2} b^{8} d^{3} x^{2} + 4 \, a^{3} b^{7} d^{3} x + a^{4} b^{6} d^{3} - {\left (a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{3} + 6 \, a^{5} b^{5} x^{2} + 4 \, a^{6} b^{4} x + a^{7} b^{3}\right )} e^{3} + 3 \, {\left (a^{2} b^{8} d x^{4} + 4 \, a^{3} b^{7} d x^{3} + 6 \, a^{4} b^{6} d x^{2} + 4 \, a^{5} b^{5} d x + a^{6} b^{4} d\right )} e^{2} - 3 \, {\left (a b^{9} d^{2} x^{4} + 4 \, a^{2} b^{8} d^{2} x^{3} + 6 \, a^{3} b^{7} d^{2} x^{2} + 4 \, a^{4} b^{6} d^{2} x + a^{5} b^{5} d^{2}\right )} e\right )}}, \frac {3 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{4} - {\left (16 \, b^{5} d^{4} + {\left (3 \, a b^{4} x^{3} + 11 \, a^{2} b^{3} x^{2} - 11 \, a^{3} b^{2} x - 3 \, a^{4} b\right )} e^{4} - {\left (3 \, b^{5} d x^{3} + 13 \, a b^{4} d x^{2} - 55 \, a^{2} b^{3} d x - a^{3} b^{2} d\right )} e^{3} + 2 \, {\left (b^{5} d^{2} x^{2} - 34 \, a b^{4} d^{2} x + 13 \, a^{2} b^{3} d^{2}\right )} e^{2} + 8 \, {\left (3 \, b^{5} d^{3} x - 5 \, a b^{4} d^{3}\right )} e\right )} \sqrt {x e + d}}{64 \, {\left (b^{10} d^{3} x^{4} + 4 \, a b^{9} d^{3} x^{3} + 6 \, a^{2} b^{8} d^{3} x^{2} + 4 \, a^{3} b^{7} d^{3} x + a^{4} b^{6} d^{3} - {\left (a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{3} + 6 \, a^{5} b^{5} x^{2} + 4 \, a^{6} b^{4} x + a^{7} b^{3}\right )} e^{3} + 3 \, {\left (a^{2} b^{8} d x^{4} + 4 \, a^{3} b^{7} d x^{3} + 6 \, a^{4} b^{6} d x^{2} + 4 \, a^{5} b^{5} d x + a^{6} b^{4} d\right )} e^{2} - 3 \, {\left (a b^{9} d^{2} x^{4} + 4 \, a^{2} b^{8} d^{2} x^{3} + 6 \, a^{3} b^{7} d^{2} x^{2} + 4 \, a^{4} b^{6} d^{2} x + a^{5} b^{5} d^{2}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.10, size = 325, normalized size = 1.20 \begin {gather*} \frac {3 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{3} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 11 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{4} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} + 3 \, \sqrt {x e + d} b^{3} d^{3} e^{4} + 11 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{5} + 22 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} - 9 \, \sqrt {x e + d} a b^{2} d^{2} e^{5} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{6} + 9 \, \sqrt {x e + d} a^{2} b d e^{6} - 3 \, \sqrt {x e + d} a^{3} e^{7}}{64 \, {\left (b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{3} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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