Optimal. Leaf size=200 \[ -\frac {35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac {21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {105 b e^3}{8 (b d-a e)^5 \sqrt {d+e x}}+\frac {105 b^{3/2} e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65,
214} \begin {gather*} \frac {105 b^{3/2} e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2}}-\frac {105 b e^3}{8 \sqrt {d+e x} (b d-a e)^5}-\frac {35 e^3}{8 (d+e x)^{3/2} (b d-a e)^4}-\frac {21 e^2}{8 (a+b x) (d+e x)^{3/2} (b d-a e)^3}+\frac {3 e}{4 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^4 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac {(3 e) \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{2 (b d-a e)}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac {\left (21 e^2\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{8 (b d-a e)^2}\\ &=-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac {21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {\left (105 e^3\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^3}\\ &=-\frac {35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac {21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {\left (105 b e^3\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^4}\\ &=-\frac {35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac {21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {105 b e^3}{8 (b d-a e)^5 \sqrt {d+e x}}-\frac {\left (105 b^2 e^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^5}\\ &=-\frac {35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac {21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {105 b e^3}{8 (b d-a e)^5 \sqrt {d+e x}}-\frac {\left (105 b^2 e^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 (b d-a e)^5}\\ &=-\frac {35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac {1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac {21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {105 b e^3}{8 (b d-a e)^5 \sqrt {d+e x}}+\frac {105 b^{3/2} e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.90, size = 220, normalized size = 1.10 \begin {gather*} \frac {1}{24} \left (\frac {-16 a^4 e^4+16 a^3 b e^3 (13 d+9 e x)+3 a^2 b^2 e^2 \left (55 d^2+318 d e x+231 e^2 x^2\right )+2 a b^3 e \left (-25 d^3+90 d^2 e x+567 d e^2 x^2+420 e^3 x^3\right )+b^4 \left (8 d^4-18 d^3 e x+63 d^2 e^2 x^2+420 d e^3 x^3+315 e^4 x^4\right )}{(-b d+a e)^5 (a+b x)^3 (d+e x)^{3/2}}+\frac {315 b^{3/2} e^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{11/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 177, normalized size = 0.88
method | result | size |
derivativedivides | \(2 e^{3} \left (-\frac {1}{3 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 b}{\left (a e -b d \right )^{5} \sqrt {e x +d}}+\frac {b^{2} \left (\frac {\frac {41 b^{2} \left (e x +d \right )^{\frac {5}{2}}}{16}+\frac {35 \left (a e -b d \right ) b \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (\frac {55}{16} a^{2} e^{2}-\frac {55}{8} a b d e +\frac {55}{16} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{5}}\right )\) | \(177\) |
default | \(2 e^{3} \left (-\frac {1}{3 \left (a e -b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 b}{\left (a e -b d \right )^{5} \sqrt {e x +d}}+\frac {b^{2} \left (\frac {\frac {41 b^{2} \left (e x +d \right )^{\frac {5}{2}}}{16}+\frac {35 \left (a e -b d \right ) b \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (\frac {55}{16} a^{2} e^{2}-\frac {55}{8} a b d e +\frac {55}{16} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{5}}\right )\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 884 vs.
\(2 (178) = 356\).
time = 3.35, size = 1779, normalized size = 8.90 \begin {gather*} \left [-\frac {315 \, {\left ({\left (b^{4} x^{5} + 3 \, a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{3} + a^{3} b x^{2}\right )} e^{5} + 2 \, {\left (b^{4} d x^{4} + 3 \, a b^{3} d x^{3} + 3 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} e^{4} + {\left (b^{4} d^{2} x^{3} + 3 \, a b^{3} d^{2} x^{2} + 3 \, a^{2} b^{2} d^{2} x + a^{3} b d^{2}\right )} e^{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 (8 \, b^{4} d^{4} + {\left (315 \, b^{4} x^{4} + 840 \, a b^{3} x^{3} + 693 \, a^{2} b^{2} x^{2} + 144 \, a^{3} b x - 16 \, a^{4}\right )} e^{4} + 2 \, {\left (210 \, b^{4} d x^{3} + 567 \, a b^{3} d x^{2} + 477 \, a^{2} b^{2} d x + 104 \, a^{3} b d\right )} e^{3} + 3 \, {\left (21 \, b^{4} d^{2} x^{2} + 60 \, a b^{3} d^{2} x + 55 \, a^{2} b^{2} d^{2}\right )} e^{2} - 2 \, {\left (9 \, b^{4} d^{3} x + 25 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{48 \, {\left (b^{8} d^{7} x^{3} + 3 \, a b^{7} d^{7} x^{2} + 3 \, a^{2} b^{6} d^{7} x + a^{3} b^{5} d^{7} - {\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )} e^{7} + {\left (5 \, a^{4} b^{4} d x^{5} + 13 \, a^{5} b^{3} d x^{4} + 9 \, a^{6} b^{2} d x^{3} - a^{7} b d x^{2} - 2 \, a^{8} d x\right )} e^{6} - {\left (10 \, a^{3} b^{5} d^{2} x^{5} + 20 \, a^{4} b^{4} d^{2} x^{4} + a^{5} b^{3} d^{2} x^{3} - 17 \, a^{6} b^{2} d^{2} x^{2} - 7 \, a^{7} b d^{2} x + a^{8} d^{2}\right )} e^{5} + 5 \, {\left (2 \, a^{2} b^{6} d^{3} x^{5} + 2 \, a^{3} b^{5} d^{3} x^{4} - 5 \, a^{4} b^{4} d^{3} x^{3} - 7 \, a^{5} b^{3} d^{3} x^{2} - a^{6} b^{2} d^{3} x + a^{7} b d^{3}\right )} e^{4} - 5 \, {\left (a b^{7} d^{4} x^{5} - a^{2} b^{6} d^{4} x^{4} - 7 \, a^{3} b^{5} d^{4} x^{3} - 5 \, a^{4} b^{4} d^{4} x^{2} + 2 \, a^{5} b^{3} d^{4} x + 2 \, a^{6} b^{2} d^{4}\right )} e^{3} + {\left (b^{8} d^{5} x^{5} - 7 \, a b^{7} d^{5} x^{4} - 17 \, a^{2} b^{6} d^{5} x^{3} + a^{3} b^{5} d^{5} x^{2} + 20 \, a^{4} b^{4} d^{5} x + 10 \, a^{5} b^{3} d^{5}\right )} e^{2} + {\left (2 \, b^{8} d^{6} x^{4} + a b^{7} d^{6} x^{3} - 9 \, a^{2} b^{6} d^{6} x^{2} - 13 \, a^{3} b^{5} d^{6} x - 5 \, a^{4} b^{4} d^{6}\right )} e\right )}}, \frac {315 \, {\left ({\left (b^{4} x^{5} + 3 \, a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{3} + a^{3} b x^{2}\right )} e^{5} + 2 \, {\left (b^{4} d x^{4} + 3 \, a b^{3} d x^{3} + 3 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} e^{4} + {\left (b^{4} d^{2} x^{3} + 3 \, a b^{3} d^{2} x^{2} + 3 \, a^{2} b^{2} d^{2} x + a^{3} b d^{2}\right )} e^{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 (8 \, b^{4} d^{4} + {\left (315 \, b^{4} x^{4} + 840 \, a b^{3} x^{3} + 693 \, a^{2} b^{2} x^{2} + 144 \, a^{3} b x - 16 \, a^{4}\right )} e^{4} + 2 \, {\left (210 \, b^{4} d x^{3} + 567 \, a b^{3} d x^{2} + 477 \, a^{2} b^{2} d x + 104 \, a^{3} b d\right )} e^{3} + 3 \, {\left (21 \, b^{4} d^{2} x^{2} + 60 \, a b^{3} d^{2} x + 55 \, a^{2} b^{2} d^{2}\right )} e^{2} - 2 \, {\left (9 \, b^{4} d^{3} x + 25 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{8} d^{7} x^{3} + 3 \, a b^{7} d^{7} x^{2} + 3 \, a^{2} b^{6} d^{7} x + a^{3} b^{5} d^{7} - {\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )} e^{7} + {\left (5 \, a^{4} b^{4} d x^{5} + 13 \, a^{5} b^{3} d x^{4} + 9 \, a^{6} b^{2} d x^{3} - a^{7} b d x^{2} - 2 \, a^{8} d x\right )} e^{6} - {\left (10 \, a^{3} b^{5} d^{2} x^{5} + 20 \, a^{4} b^{4} d^{2} x^{4} + a^{5} b^{3} d^{2} x^{3} - 17 \, a^{6} b^{2} d^{2} x^{2} - 7 \, a^{7} b d^{2} x + a^{8} d^{2}\right )} e^{5} + 5 \, {\left (2 \, a^{2} b^{6} d^{3} x^{5} + 2 \, a^{3} b^{5} d^{3} x^{4} - 5 \, a^{4} b^{4} d^{3} x^{3} - 7 \, a^{5} b^{3} d^{3} x^{2} - a^{6} b^{2} d^{3} x + a^{7} b d^{3}\right )} e^{4} - 5 \, {\left (a b^{7} d^{4} x^{5} - a^{2} b^{6} d^{4} x^{4} - 7 \, a^{3} b^{5} d^{4} x^{3} - 5 \, a^{4} b^{4} d^{4} x^{2} + 2 \, a^{5} b^{3} d^{4} x + 2 \, a^{6} b^{2} d^{4}\right )} e^{3} + {\left (b^{8} d^{5} x^{5} - 7 \, a b^{7} d^{5} x^{4} - 17 \, a^{2} b^{6} d^{5} x^{3} + a^{3} b^{5} d^{5} x^{2} + 20 \, a^{4} b^{4} d^{5} x + 10 \, a^{5} b^{3} d^{5}\right )} e^{2} + {\left (2 \, b^{8} d^{6} x^{4} + a b^{7} d^{6} x^{3} - 9 \, a^{2} b^{6} d^{6} x^{2} - 13 \, a^{3} b^{5} d^{6} x - 5 \, a^{4} b^{4} d^{6}\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 (a + b x\right )^{4} \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 427 vs.
\(2 (178) = 356\).
time = 1.35, size = 427, normalized size = 2.14 \begin {gather*} -\frac {105 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e}} - \frac {315 \, {\left (x e + d\right )}^{4} b^{4} e^{3} - 840 \, {\left (x e + d\right )}^{3} b^{4} d e^{3} + 693 \, {\left (x e + d\right )}^{2} b^{4} d^{2} e^{3} - 144 \, {\left (x e + d\right )} b^{4} d^{3} e^{3} - 16 \, b^{4} d^{4} e^{3} + 840 \, {\left (x e + d\right )}^{3} a b^{3} e^{4} - 1386 \, {\left (x e + d\right )}^{2} a b^{3} d e^{4} + 432 \, {\left (x e + d\right )} a b^{3} d^{2} e^{4} + 64 \, a b^{3} d^{3} e^{4} + 693 \, {\left (x e + d\right )}^{2} a^{2} b^{2} e^{5} - 432 \, {\left (x e + d\right )} a^{2} b^{2} d e^{5} - 96 \, a^{2} b^{2} d^{2} e^{5} + 144 \, {\left (x e + d\right )} a^{3} b e^{6} + 64 \, a^{3} b d e^{6} - 16 \, a^{4} e^{7}}{24 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} b - \sqrt {x e + d} b d + \sqrt {x e + d} a e\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.87, size = 334, normalized size = 1.67 \begin {gather*} \frac {\frac {231\,b^2\,e^3\,{\left (d+e\,x\right )}^2}{8\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^3}{3\,\left (a\,e-b\,d\right )}+\frac {35\,b^3\,e^3\,{\left (d+e\,x\right )}^3}{{\left (a\,e-b\,d\right )}^4}+\frac {105\,b^4\,e^3\,{\left (d+e\,x\right )}^4}{8\,{\left (a\,e-b\,d\right )}^5}+\frac {6\,b\,e^3\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )+b^3\,{\left (d+e\,x\right )}^{9/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}+{\left (d+e\,x\right )}^{5/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}+\frac {105\,b^{3/2}\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^{11/2}}\right )}{8\,{\left (a\,e-b\,d\right )}^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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