Optimal. Leaf size=112 \[ \frac {2 (B d-A e) \left (c d^2+a e^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {2 \left (3 B c d^2-2 A c d e+a B e^2\right )}{e^4 \sqrt {d+e x}}-\frac {2 c (3 B d-A e) \sqrt {d+e x}}{e^4}+\frac {2 B c (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {786}
\begin {gather*} -\frac {2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac {2 c \sqrt {d+e x} (3 B d-A e)}{e^4}+\frac {2 B c (d+e x)^{3/2}}{3 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 786
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^{5/2}}+\frac {3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^{3/2}}+\frac {c (-3 B d+A e)}{e^3 \sqrt {d+e x}}+\frac {B c \sqrt {d+e x}}{e^3}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {2 \left (3 B c d^2-2 A c d e+a B e^2\right )}{e^4 \sqrt {d+e x}}-\frac {2 c (3 B d-A e) \sqrt {d+e x}}{e^4}+\frac {2 B c (d+e x)^{3/2}}{3 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 94, normalized size = 0.84 \begin {gather*} -\frac {2 \left (a A e^3+a B e^2 (2 d+3 e x)-A c e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B c \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 108, normalized size = 0.96
method | result | size |
risch | \(\frac {2 c \left (B e x +3 A e -8 B d \right ) \sqrt {e x +d}}{3 e^{4}}-\frac {2 \left (-6 A c d \,e^{2} x +3 B a \,e^{3} x +9 B c \,d^{2} x e +A a \,e^{3}-5 A c \,d^{2} e +2 a B d \,e^{2}+8 B c \,d^{3}\right )}{3 e^{4} \left (e x +d \right )^{\frac {3}{2}}}\) | \(96\) |
gosper | \(-\frac {2 \left (-B c \,x^{3} e^{3}-3 A c \,e^{3} x^{2}+6 B c d \,e^{2} x^{2}-12 A c d \,e^{2} x +3 B a \,e^{3} x +24 B c \,d^{2} x e +A a \,e^{3}-8 A c \,d^{2} e +2 a B d \,e^{2}+16 B c \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(100\) |
trager | \(-\frac {2 \left (-B c \,x^{3} e^{3}-3 A c \,e^{3} x^{2}+6 B c d \,e^{2} x^{2}-12 A c d \,e^{2} x +3 B a \,e^{3} x +24 B c \,d^{2} x e +A a \,e^{3}-8 A c \,d^{2} e +2 a B d \,e^{2}+16 B c \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}}\) | \(100\) |
derivativedivides | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A c e \sqrt {e x +d}-6 B c d \sqrt {e x +d}-\frac {2 \left (-2 A c d e +B \,e^{2} a +3 B c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (A a \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(108\) |
default | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A c e \sqrt {e x +d}-6 B c d \sqrt {e x +d}-\frac {2 \left (-2 A c d e +B \,e^{2} a +3 B c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (A a \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 109, normalized size = 0.97 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} B c - 3 \, {\left (3 \, B c d - A c e\right )} \sqrt {x e + d}\right )} e^{\left (-3\right )} + \frac {{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3} - 3 \, {\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} {\left (x e + d\right )}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.21, size = 109, normalized size = 0.97 \begin {gather*} -\frac {2 \, {\left (16 \, B c d^{3} - {\left (B c x^{3} + 3 \, A c x^{2} - 3 \, B a x - A a\right )} e^{3} + 2 \, {\left (3 \, B c d x^{2} - 6 \, A c d x + B a d\right )} e^{2} + 8 \, {\left (3 \, B c d^{2} x - A c d^{2}\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 449 vs.
\(2 (114) = 228\).
time = 0.46, size = 449, normalized size = 4.01 \begin {gather*} \begin {cases} - \frac {2 A a e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {16 A c d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {24 A c d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {6 A c e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {4 B a d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 B a e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 B c d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 B c d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 B c d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 B c e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a x + \frac {A c x^{3}}{3} + \frac {B a x^{2}}{2} + \frac {B c x^{4}}{4}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 126, normalized size = 1.12 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B c e^{8} - 9 \, \sqrt {x e + d} B c d e^{8} + 3 \, \sqrt {x e + d} A c e^{9}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )} B c d^{2} - B c d^{3} - 6 \, {\left (x e + d\right )} A c d e + A c d^{2} e + 3 \, {\left (x e + d\right )} B a e^{2} - B a d e^{2} + A a e^{3}\right )} e^{\left (-4\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.77, size = 113, normalized size = 1.01 \begin {gather*} \frac {2\,B\,c\,{\left (d+e\,x\right )}^3-2\,A\,a\,e^3+2\,B\,c\,d^3+2\,B\,a\,d\,e^2-2\,A\,c\,d^2\,e-6\,B\,a\,e^2\,\left (d+e\,x\right )+6\,A\,c\,e\,{\left (d+e\,x\right )}^2-18\,B\,c\,d\,{\left (d+e\,x\right )}^2-18\,B\,c\,d^2\,\left (d+e\,x\right )+12\,A\,c\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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