Optimal. Leaf size=161 \[ \frac {2 (a+b x) \sqrt {d+e x} (b d-a e)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {646, 50, 63, 208} \begin {gather*} \frac {2 (a+b x) \sqrt {d+e x} (b d-a e)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) (b d-a e)^{3/2} \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
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e) (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e) (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^2 \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^4 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (b d-a e) (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{3/2} (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 [A] time = 0.05, size = 96, normalized size = 0.60 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )}{3 b^{5/2} \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 14.09, size = 122, normalized size = 0.76 \begin {gather*} \frac {(-a e-b e x) \left (\frac {2 (a e-b d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{5/2}}-\frac {2 \sqrt {d+e x} (-3 a e+b (d+e x)+3 b d)}{3 b^2}\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 = 0.42, size = 188, normalized size = 1.17 \begin {gather*} \left [-\frac {3 \, {\left (b d - 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 (b e x + 4 \, b d - 3 \, a e\right )} \sqrt {e x + d}}{3 \, b^{2}}, -\frac {2 \, {\left (3 \, {\left (b d - 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 (b e x + 4 \, b d - 3 \, a e\right )} \sqrt {e x + d}\right )}}{3 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 148, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{2}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {x e + d} b^{2} d \mathrm {sgn}\left (b x + a\right ) - 3 \, \sqrt {x e + d} a b e \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 188, normalized size = 1.17 \begin {gather*} \frac {2 \left (b x +a \right ) \left (3 a^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-6 a b d e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+3 b^{2} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-3 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a e +3 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b d +\left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b \right )}{3 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {\left (a e -b d \right ) b}\, 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 (e x + d\right )}^{\frac {3}{2}}}{\sqrt {{\left (b x + a\right )}^{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 (d+e\,x\right )}^{3/2}}{\sqrt {{\left (a+b\,x\right )}^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 (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (a + b x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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