Optimal. Leaf size=148 \[ -\frac {(d+e x)^{3/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 148, 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 = {768, 646, 50, 63, 208} \begin {gather*} -\frac {(d+e x)^{3/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 646
Rule 768
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {(d+e x)^{3/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 e) \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=-\frac {(d+e x)^{3/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 e (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 e (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 \left (b^2 d-a b e\right ) \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 )}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {3 e (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{3/2}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 66, normalized size = 0.45 \begin {gather*} \frac {2 e (a+b x) (d+e x)^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};-\frac {b (d+e x)}{a e-b d}\right )}{5 \sqrt {(a+b x)^2} (a e-b d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 21.40, size = 140, normalized size = 0.95 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {3 e \sqrt {a e-b d} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{5/2}}-\frac {e \sqrt {d+e x} (3 a e+2 b (d+e x)-3 b d)}{b^2 (a e+b (d+e x)-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 [A] time = 0.43, size = 210, normalized size = 1.42 \begin {gather*} \left [\frac {3 \, {\left (b e x + a e\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (2 \, b e x - b d + 3 \, a e\right )} \sqrt {e x + d}}{2 \, {\left (b^{3} x + a b^{2}\right )}}, -\frac {3 \, {\left (b e x + a e\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (2 \, b e x - b d + 3 \, a e\right )} \sqrt {e x + d}}{b^{3} x + a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 198, normalized size = 1.34 \begin {gather*} \frac {3 \, {\left (b d e^{2} - a e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{\sqrt {-b^{2} d + a b e} b^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, \sqrt {x e + d} e}{b^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {{\left (\sqrt {x e + d} b d e^{2} - \sqrt {x e + d} a e^{3}\right )} e^{\left (-1\right )}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 222, normalized size = 1.50 \begin {gather*} \frac {\left (-3 a b \,e^{2} x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+3 b^{2} d e x \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-3 a^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+3 a b d e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+2 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b e x +3 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a e -\sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b d \right ) \left (b x +a \right )^{2}}{\sqrt {\left (a e -b d \right ) b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{2}} \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 )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\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 ) \left (d + e x\right )^{\frac {3}{2}}}{\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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