Optimal. Leaf size=223 \[ \frac {5 e}{4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^2 (a+b x)}{4 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 \sqrt {b} e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {15 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)^3}-\frac {15 \sqrt {b} 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)^{7/2}}+\frac {5 e}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}-\frac {1}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)} \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)^{3/2} \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)^{3/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{2 (b d-a e) (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 b e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{3/2}} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e}{4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (15 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)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e}{4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^2 (a+b x)}{4 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (15 b 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)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e}{4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^2 (a+b x)}{4 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (15 b 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)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 e}{4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^2 (a+b x)}{4 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 \sqrt {b} 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)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 155, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {-b d+a e} \left (8 a^2 e^2+a b e (9 d+25 e x)+b^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )\right )+15 \sqrt {b} e^2 (a+b x)^2 \sqrt {d+e x} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 (-b d+a e)^{7/2} (a+b x) \sqrt {(a+b x)^2} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 285, normalized size = 1.28
method | result | size |
default | \(-\frac {\left (15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \sqrt {e x +d}\, b^{3} e^{2} x^{2}+30 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \sqrt {e x +d}\, a \,b^{2} e^{2} x +15 \sqrt {b \left (a e -b d \right )}\, b^{2} e^{2} x^{2}+15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \sqrt {e x +d}\, a^{2} b \,e^{2}+25 \sqrt {b \left (a e -b d \right )}\, a b \,e^{2} x +5 \sqrt {b \left (a e -b d \right )}\, b^{2} d e x +8 \sqrt {b \left (a e -b d \right )}\, a^{2} e^{2}+9 \sqrt {b \left (a e -b d \right )}\, a b d e -2 \sqrt {b \left (a e -b d \right )}\, b^{2} d^{2}\right ) \left (b x +a \right )}{4 \sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right )^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(285\) |
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. 375 vs.
\(2 (163) = 326\).
time = 1.45, size = 762, normalized size = 3.42 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} e^{3} + {\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d\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 (2 \, b^{2} d^{2} - {\left (15 \, b^{2} x^{2} + 25 \, a b x + 8 \, a^{2}\right )} e^{2} - {\left (5 \, b^{2} d x + 9 \, a b d\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{5} d^{4} x^{2} + 2 \, a b^{4} d^{4} x + a^{2} b^{3} d^{4} - {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )} e^{4} + {\left (3 \, a^{2} b^{3} d x^{3} + 5 \, a^{3} b^{2} d x^{2} + a^{4} b d x - a^{5} d\right )} e^{3} - 3 \, {\left (a b^{4} d^{2} x^{3} + a^{2} b^{3} d^{2} x^{2} - a^{3} b^{2} d^{2} x - a^{4} b d^{2}\right )} e^{2} + {\left (b^{5} d^{3} x^{3} - a b^{4} d^{3} x^{2} - 5 \, a^{2} b^{3} d^{3} x - 3 \, a^{3} b^{2} d^{3}\right )} e\right )}}, -\frac {15 \, {\left ({\left (b^{2} x^{3} + 2 \, a b x^{2} + a^{2} x\right )} e^{3} + {\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d\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 (2 \, b^{2} d^{2} - {\left (15 \, b^{2} x^{2} + 25 \, a b x + 8 \, a^{2}\right )} e^{2} - {\left (5 \, b^{2} d x + 9 \, a b d\right )} e\right )} \sqrt {x e + d}}{4 \, {\left (b^{5} d^{4} x^{2} + 2 \, a b^{4} d^{4} x + a^{2} b^{3} d^{4} - {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )} e^{4} + {\left (3 \, a^{2} b^{3} d x^{3} + 5 \, a^{3} b^{2} d x^{2} + a^{4} b d x - a^{5} d\right )} e^{3} - 3 \, {\left (a b^{4} d^{2} x^{3} + a^{2} b^{3} d^{2} x^{2} - a^{3} b^{2} d^{2} x - a^{4} b d^{2}\right )} e^{2} + {\left (b^{5} d^{3} x^{3} - a b^{4} d^{3} x^{2} - 5 \, a^{2} b^{3} d^{3} x - 3 \, a^{3} b^{2} d^{3}\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 {3}{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 = 0.97, size = 307, normalized size = 1.38 \begin {gather*} \frac {15 \, b \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, e^{2}}{{\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {x e + d}} + \frac {7 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} e^{2} - 9 \, \sqrt {x e + d} b^{2} d e^{2} + 9 \, \sqrt {x e + d} a b e^{3}}{4 \, {\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \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 )}^{3/2}\,{\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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