Optimal. Leaf size=215 \[ -\frac {(d+e x)^{3/2} (A b-a B)}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {\sqrt {d+e x} (-3 a B e-A b e+4 b B d)}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {e (a+b x) (-3 a B e-A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]
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Rubi [A] time = 0.19, 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 = {770, 78, 47, 63, 208} \begin {gather*} -\frac {(d+e x)^{3/2} (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} (-3 a B e-A b e+4 b B d)}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {e (a+b x) (-3 a B e-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^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 208
Rule 770
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) \sqrt {d+e x}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{3/2}}{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-A b e-3 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^2} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d-A b e-3 a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{3/2}}{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-A b e-3 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^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d-A b e-3 a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{3/2}}{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-A b e-3 a B e) \left (a b+b^2 x\right )\right ) \operatorname {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^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(4 b B d-A b e-3 a B e) \sqrt {d+e x}}{4 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{3/2}}{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-A b e-3 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{5/2} (b d-a e)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 167, normalized size = 0.78 \begin {gather*} \frac {(a+b x) \left (\frac {(a+b x) (-3 a B e-A b e+4 b B d) \left (\sqrt {b} e (a+b x) \sqrt {d+e x} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )-b (d+e x) \sqrt {a e-b d}\right )}{\sqrt {a e-b d}}-2 b^2 (d+e x)^2 (A b-a B)\right )}{4 b^3 \left ((a+b x)^2\right )^{3/2} \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 42.64, size = 243, normalized size = 1.13 \begin {gather*} \frac {(-a e-b e x) \left (\frac {e \sqrt {d+e x} \left (-3 a^2 B e^2-a A b e^2-5 a b B e (d+e x)+7 a b B d e+A b^2 e (d+e x)+A b^2 d e-4 b^2 B d^2+4 b^2 B d (d+e x)\right )}{4 b^2 (b d-a e) (-a e-b (d+e x)+b d)^2}+\frac {\left (-3 a B e^2-A b e^2+4 b B d e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 b^{5/2} (b d-a e) \sqrt {a e-b d}}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 721, normalized size = 3.35 \begin {gather*} \left [\frac {{\left (4 \, B a^{2} b d e - {\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (5 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e + {\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2} + {\left (4 \, B b^{4} d^{2} - {\left (9 \, B a b^{3} - A b^{4}\right )} d e + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{5} d^{2} - 2 \, a^{3} b^{4} d e + a^{4} b^{3} e^{2} + {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} x^{2} + 2 \, {\left (a b^{6} d^{2} - 2 \, a^{2} b^{5} d e + a^{3} b^{4} e^{2}\right )} x\right )}}, \frac {{\left (4 \, B a^{2} b d e - {\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (5 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e + {\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2} + {\left (4 \, B b^{4} d^{2} - {\left (9 \, B a b^{3} - A b^{4}\right )} d e + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{5} d^{2} - 2 \, a^{3} b^{4} d e + a^{4} b^{3} e^{2} + {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} x^{2} + 2 \, {\left (a b^{6} d^{2} - 2 \, a^{2} b^{5} d e + a^{3} b^{4} e^{2}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 337, normalized size = 1.57 \begin {gather*} \frac {{\left (4 \, B b d e^{2} - 3 \, 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 )}{4 \, {\left (b^{3} d e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a b^{2} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} d e^{2} - 4 \, \sqrt {x e + d} B b^{2} d^{2} e^{2} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b e^{3} + {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} e^{3} + 7 \, \sqrt {x e + d} B a b d e^{3} + \sqrt {x e + d} A b^{2} d e^{3} - 3 \, \sqrt {x e + d} B a^{2} e^{4} - \sqrt {x e + d} A a b e^{4}}{4 \, {\left (b^{3} d e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a b^{2} e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\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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maple [B] time = 0.07, size = 555, normalized size = 2.58 \begin {gather*} \frac {\left (A \,b^{3} e^{3} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+3 B a \,b^{2} e^{3} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-4 B \,b^{3} d \,e^{2} x^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+2 A a \,b^{2} e^{3} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+6 B \,a^{2} b \,e^{3} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-8 B a \,b^{2} d \,e^{2} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+A \,a^{2} b \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+3 B \,a^{3} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-4 B \,a^{2} b d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-\sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, A a b \,e^{2}+\sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, A \,b^{2} d e -3 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, B \,a^{2} e^{2}+7 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, B a b d e -4 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, B \,b^{2} d^{2}+\sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} A \,b^{2} e -5 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} B a b e +4 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} B \,b^{2} d \right ) \left (b x +a \right )}{4 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{2} e} \end {gather*}
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*} \int \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \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 {\left (A+B\,x\right )\,\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \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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