Optimal. Leaf size=215 \[ -\frac {(4 b B d-3 A b e-a B e) \sqrt {d+e x}}{4 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (4 b B d-3 A b e-a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {784, 79, 44, 65,
214} \begin {gather*} -\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {\sqrt {d+e x} (-a B e-3 A b e+4 b B d)}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac {e (a+b x) (-a B e-3 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 79
Rule 214
Rule 784
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 )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((4 b B d-3 A b e-a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d-3 A b e-a B e) \sqrt {d+e x}}{4 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (e (4 b B d-3 A b e-a B e) \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 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d-3 A b e-a B e) \sqrt {d+e x}}{4 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left ((4 b B d-3 A b e-a 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 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d-3 A b e-a B e) \sqrt {d+e x}}{4 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (4 b B d-3 A b e-a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 167, normalized size = 0.78 \begin {gather*} \frac {e (a+b x)^3 \left (\frac {\sqrt {b} \sqrt {d+e x} \left (A b (-2 b d+5 a e+3 b e x)-B \left (a^2 e+4 b^2 d x+a b (2 d-e x)\right )\right )}{e (b d-a e)^2 (a+b x)^2}-\frac {(4 b B d-3 A b e-a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}}\right )}{4 b^{3/2} \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(555\) vs.
\(2(162)=324\).
time = 0.99, size = 556, normalized size = 2.59
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (3 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{3} e^{3} x^{2}+B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{2} e^{3} x^{2}-4 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{3} d \,e^{2} x^{2}+6 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{2} e^{3} x +2 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b \,e^{3} x -8 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{2} d \,e^{2} x +3 A \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{2} e +3 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b \,e^{3}+B \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a b e -4 B \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{2} d +B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} e^{3}-4 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b d \,e^{2}+5 A \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a b \,e^{2}-5 A \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{2} d e -B \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} e^{2}-3 B \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a b d e +4 B \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{2} d^{2}\right )}{4 e \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right )^{2} b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(556\) |
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. 387 vs.
\(2 (174) = 348\).
time = 4.21, size = 788, normalized size = 3.67 \begin {gather*} \left [\frac {\sqrt {b^{2} d - a b e} {\left ({\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} e^{2} - 4 \, {\left (B b^{3} d x^{2} + 2 \, B a b^{2} d x + B a^{2} b d\right )} e\right )} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) - 2 \, {\left (4 \, B b^{4} d^{2} x + 2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} e^{2} - {\left ({\left (5 \, B a b^{3} + 3 \, A b^{4}\right )} d x + {\left (B a^{2} b^{2} + 7 \, A a b^{3}\right )} d\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{7} d^{3} x^{2} + 2 \, a b^{6} d^{3} x + a^{2} b^{5} d^{3} - {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )} e^{3} + 3 \, {\left (a^{2} b^{5} d x^{2} + 2 \, a^{3} b^{4} d x + a^{4} b^{3} d\right )} e^{2} - 3 \, {\left (a b^{6} d^{2} x^{2} + 2 \, a^{2} b^{5} d^{2} x + a^{3} b^{4} d^{2}\right )} e\right )}}, \frac {\sqrt {-b^{2} d + a b e} {\left ({\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} e^{2} - 4 \, {\left (B b^{3} d x^{2} + 2 \, B a b^{2} d x + B a^{2} b d\right )} e\right )} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) - {\left (4 \, B b^{4} d^{2} x + 2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} e^{2} - {\left ({\left (5 \, B a b^{3} + 3 \, A b^{4}\right )} d x + {\left (B a^{2} b^{2} + 7 \, A a b^{3}\right )} d\right )} e\right )} \sqrt {x e + d}}{4 \, {\left (b^{7} d^{3} x^{2} + 2 \, a b^{6} d^{3} x + a^{2} b^{5} d^{3} - {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )} e^{3} + 3 \, {\left (a^{2} b^{5} d x^{2} + 2 \, a^{3} b^{4} d x + a^{4} b^{3} d\right )} e^{2} - 3 \, {\left (a b^{6} d^{2} x^{2} + 2 \, a^{2} b^{5} d^{2} x + a^{3} b^{4} d^{2}\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 {A + B x}{\sqrt {d + e x} \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 = 0.96, size = 302, normalized size = 1.40 \begin {gather*} -\frac {{\left (4 \, B b d e - B a e^{2} - 3 \, A b e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} d e - 4 \, \sqrt {x e + d} B b^{2} d^{2} e - {\left (x e + d\right )}^{\frac {3}{2}} B a b e^{2} - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} e^{2} + 3 \, \sqrt {x e + d} B a b d e^{2} + 5 \, \sqrt {x e + d} A b^{2} d e^{2} + \sqrt {x e + d} B a^{2} e^{3} - 5 \, \sqrt {x e + d} A a b e^{3}}{4 \, {\left (b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b e^{2} \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 {A+B\,x}{\sqrt {d+e\,x}\,{\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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