Optimal. Leaf size=41 \[ \frac {2 (a+b x) (d+e x)^{7/2}}{7 e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 32} \begin {gather*} \frac {2 (a+b x) (d+e x)^{7/2}}{7 e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(a+b x) (d+e x)^{5/2}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int (d+e x)^{5/2} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) (d+e x)^{7/2}}{7 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.78 \begin {gather*} \frac {2 (a+b x) (d+e x)^{7/2}}{7 e \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 35.02, size = 44, normalized size = 1.07 \begin {gather*} -\frac {2 (d+e x)^{7/2} (-a e-b e x)}{7 e^2 \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 39, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {e x + d}}{7 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 154, normalized size = 3.76 \begin {gather*} \frac {2}{35} \, {\left (35 \, \sqrt {x e + d} d^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} d^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} d \mathrm {sgn}\left (b x + a\right ) + {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.66 \begin {gather*} \frac {2 \left (b x +a \right ) \left (e x +d \right )^{\frac {7}{2}}}{7 \sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 39, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {e x + d}}{7 \, e} \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 = -0.02 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{\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(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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