Optimal. Leaf size=209 \[ \frac {e^2 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{5/2}}-\frac {e \sqrt {d+e x} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x) (b d-a e)^2}-\frac {\sqrt {d+e x} (-a B e-A b e+2 b B d)}{4 b^2 (a+b x)^2 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.18, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 51, 63, 208} \[ \frac {e^2 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{5/2}}-\frac {e \sqrt {d+e x} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x) (b d-a e)^2}-\frac {\sqrt {d+e x} (-a B e-A b e+2 b B d)}{4 b^2 (a+b x)^2 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^4} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(2 b B d-A b e-a B e) \int \frac {\sqrt {d+e x}}{(a+b x)^3} \, dx}{2 b (b d-a e)}\\ &=-\frac {(2 b B d-A b e-a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(e (2 b B d-A b e-a B e)) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 b^2 (b d-a e)}\\ &=-\frac {(2 b B d-A b e-a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{8 b^2 (b d-a e)^2 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^3}-\frac {\left (e^2 (2 b B d-A b e-a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^2 (b d-a e)^2}\\ &=-\frac {(2 b B d-A b e-a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{8 b^2 (b d-a e)^2 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^3}-\frac {(e (2 b B d-A b e-a B e)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^2 (b d-a e)^2}\\ &=-\frac {(2 b B d-A b e-a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {e (2 b B d-A b e-a B e) \sqrt {d+e x}}{8 b^2 (b d-a e)^2 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^3}+\frac {e^2 (2 b B d-A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 98, normalized size = 0.47 \[ \frac {(d+e x)^{3/2} \left (\frac {3 e^2 (a B e+A b e-2 b B d) \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}+\frac {3 a B-3 A b}{(a+b x)^3}\right )}{9 b (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 1218, normalized size = 5.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 386, normalized size = 1.85 \[ -\frac {{\left (2 \, B b d e^{2} - B a e^{3} - A b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {6 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} d e^{2} - 6 \, \sqrt {x e + d} B b^{3} d^{3} e^{2} - 3 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{2} e^{3} - 3 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} e^{3} - 8 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} + 8 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} + 15 \, \sqrt {x e + d} B a b^{2} d^{2} e^{3} + 3 \, \sqrt {x e + d} A b^{3} d^{2} e^{3} + 8 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} - 8 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} - 12 \, \sqrt {x e + d} B a^{2} b d e^{4} - 6 \, \sqrt {x e + d} A a b^{2} d e^{4} + 3 \, \sqrt {x e + d} B a^{3} e^{5} + 3 \, \sqrt {x e + d} A a^{2} b e^{5}}{24 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 494, normalized size = 2.36 \[ \frac {\left (e x +d \right )^{\frac {5}{2}} A b \,e^{3}}{8 \left (b e x +a e \right )^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {\left (e x +d \right )^{\frac {5}{2}} B a \,e^{3}}{8 \left (b e x +a e \right )^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {\left (e x +d \right )^{\frac {5}{2}} B b d \,e^{2}}{4 \left (b e x +a e \right )^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {A \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b}+\frac {\left (e x +d \right )^{\frac {3}{2}} A \,e^{3}}{3 \left (b e x +a e \right )^{3} \left (a e -b d \right )}-\frac {\left (e x +d \right )^{\frac {3}{2}} B a \,e^{3}}{3 \left (b e x +a e \right )^{3} \left (a e -b d \right ) b}+\frac {B a \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {B d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b}-\frac {\sqrt {e x +d}\, A \,e^{3}}{8 \left (b e x +a e \right )^{3} b}-\frac {\sqrt {e x +d}\, B a \,e^{3}}{8 \left (b e x +a e \right )^{3} b^{2}}+\frac {\sqrt {e x +d}\, B d \,e^{2}}{4 \left (b e x +a e \right )^{3} b} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 310, normalized size = 1.48 \[ \frac {\frac {{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e^3+B\,a\,e^3-2\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^2}-\frac {\sqrt {d+e\,x}\,\left (A\,b\,e^3+B\,a\,e^3-2\,B\,b\,d\,e^2\right )}{8\,b^2}+\frac {\left (A\,b\,e^3-B\,a\,e^3\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2}+\frac {e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^3+B\,a\,e^3-2\,B\,b\,d\,e^2\right )}\right )\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{8\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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