Optimal. Leaf size=219 \[ \frac {2 (a+b x)}{5 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (a+b x)}{(b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 53, 65,
214} \begin {gather*} \frac {2 b^2 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}+\frac {2 b (a+b x)}{3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}+\frac {2 (a+b x)}{5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac {2 b^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x)}{5 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x)}{5 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^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}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x)}{5 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (a+b x)}{(b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x)}{5 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (a+b x)}{(b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 b^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 )}{e (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x)}{5 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b (a+b x)}{3 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b^2 (a+b x)}{(b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(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 = 0.25, size = 133, normalized size = 0.61 \begin {gather*} \frac {2 (a+b x) \left (\frac {3 a^2 e^2-a b e (11 d+5 e x)+b^2 \left (23 d^2+35 d e x+15 e^2 x^2\right )}{(b d-a e)^3 (d+e x)^{5/2}}-\frac {15 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}}\right )}{15 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 202, normalized size = 0.92
method | result | size |
default | \(-\frac {2 \left (b x +a \right ) \left (15 b^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}}+15 \sqrt {b \left (a e -b d \right )}\, b^{2} e^{2} x^{2}-5 \sqrt {b \left (a e -b d \right )}\, a b \,e^{2} x +35 \sqrt {b \left (a e -b d \right )}\, b^{2} d e x +3 \sqrt {b \left (a e -b d \right )}\, a^{2} e^{2}-11 \sqrt {b \left (a e -b d \right )}\, a b d e +23 \sqrt {b \left (a e -b d \right )}\, b^{2} d^{2}\right )}{15 \sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {5}{2}} \sqrt {b \left (a e -b d \right )}}\) | \(202\) |
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. 342 vs.
\(2 (164) = 328\).
time = 3.31, size = 695, normalized size = 3.17 \begin {gather*} \left [-\frac {15 \, {\left (b^{2} x^{3} e^{3} + 3 \, b^{2} d x^{2} e^{2} + 3 \, b^{2} d^{2} x e + b^{2} d^{3}\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 (23 \, b^{2} d^{2} + {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )} e^{2} + {\left (35 \, b^{2} d x - 11 \, a b d\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (b^{3} d^{6} - a^{3} x^{3} e^{6} + 3 \, {\left (a^{2} b d x^{3} - a^{3} d x^{2}\right )} e^{5} - 3 \, {\left (a b^{2} d^{2} x^{3} - 3 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} x\right )} e^{4} + {\left (b^{3} d^{3} x^{3} - 9 \, a b^{2} d^{3} x^{2} + 9 \, a^{2} b d^{3} x - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (b^{3} d^{4} x^{2} - 3 \, a b^{2} d^{4} x + a^{2} b d^{4}\right )} e^{2} + 3 \, {\left (b^{3} d^{5} x - a b^{2} d^{5}\right )} e\right )}}, -\frac {2 \, {\left (15 \, {\left (b^{2} x^{3} e^{3} + 3 \, b^{2} d x^{2} e^{2} + 3 \, b^{2} d^{2} x e + b^{2} d^{3}\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 (23 \, b^{2} d^{2} + {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )} e^{2} + {\left (35 \, b^{2} d x - 11 \, a b d\right )} e\right )} \sqrt {x e + d}\right )}}{15 \, {\left (b^{3} d^{6} - a^{3} x^{3} e^{6} + 3 \, {\left (a^{2} b d x^{3} - a^{3} d x^{2}\right )} e^{5} - 3 \, {\left (a b^{2} d^{2} x^{3} - 3 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} x\right )} e^{4} + {\left (b^{3} d^{3} x^{3} - 9 \, a b^{2} d^{3} x^{2} + 9 \, a^{2} b d^{3} x - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (b^{3} d^{4} x^{2} - 3 \, a b^{2} d^{4} x + a^{2} b d^{4}\right )} e^{2} + 3 \, {\left (b^{3} d^{5} x - a b^{2} d^{5}\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 = 1.05, size = 196, normalized size = 0.89 \begin {gather*} \frac {2}{15} \, {\left (\frac {15 \, b^{3} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {15 \, {\left (x e + d\right )}^{2} b^{2} + 5 \, {\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} - 5 \, {\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {5}{2}}}\right )} \mathrm {sgn}\left (b x + a\right ) \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}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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