Optimal. Leaf size=172 \[ \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{2 b^4}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)^3}{4 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4} \]
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Rubi [A] time = 0.13, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \begin {gather*} \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{2 b^4}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)^3}{4 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (d+e x)^3 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(b d-a e)^3 \left (a b+b^2 x\right )^3}{b^3}+\frac {3 e (b d-a e)^2 \left (a b+b^2 x\right )^4}{b^4}+\frac {3 e^2 (b d-a e) \left (a b+b^2 x\right )^5}{b^5}+\frac {e^3 \left (a b+b^2 x\right )^6}{b^6}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^4}+\frac {3 e (b d-a e)^2 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^4}+\frac {e^2 (b d-a e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^4}+\frac {e^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 171, normalized size = 0.99 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (35 a^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+21 a^2 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+7 a b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )}{140 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.86, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 167, normalized size = 0.97 \begin {gather*} \frac {1}{7} \, b^{3} e^{3} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} + {\left (a b^{2} d^{3} + 3 \, a^{2} b d^{2} e + a^{3} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 280, normalized size = 1.63 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{3} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, b^{3} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{5} \, a b^{2} d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, a b^{2} d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a^{2} b x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, a^{2} b d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{3} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{3} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{3} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{3} d^{3} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 206, normalized size = 1.20 \begin {gather*} \frac {\left (20 b^{3} e^{3} x^{6}+70 x^{5} a \,b^{2} e^{3}+70 x^{5} b^{3} d \,e^{2}+84 x^{4} a^{2} b \,e^{3}+252 x^{4} a \,b^{2} d \,e^{2}+84 x^{4} b^{3} d^{2} e +35 x^{3} a^{3} e^{3}+315 x^{3} a^{2} b d \,e^{2}+315 x^{3} a \,b^{2} d^{2} e +35 x^{3} b^{3} d^{3}+140 a^{3} d \,e^{2} x^{2}+420 a^{2} b \,d^{2} e \,x^{2}+140 a \,b^{2} d^{3} x^{2}+210 x \,a^{3} d^{2} e +210 x \,a^{2} b \,d^{3}+140 a^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{140 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.13, size = 400, normalized size = 2.33 \begin {gather*} \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{3} x - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{2} e x}{4 \, b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d e^{2} x}{4 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e^{3} x}{4 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{3} x^{2}}{7 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{3}}{4 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{2} e}{4 \, b^{2}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} d e^{2}}{4 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} e^{3}}{4 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d e^{2} x}{2 \, b^{2}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{3} x}{14 \, b^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{2} e}{5 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d e^{2}}{10 \, b^{3}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{3}}{70 \, b^{4}} \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 {\left (d+e\,x\right )}^3\,{\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 \left (d + e x\right )^{3} \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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