Optimal. Leaf size=219 \[ \frac {4 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac {3 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac {4 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \]
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Rubi [A] time = 0.09, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {645} \begin {gather*} \frac {4 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac {3 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac {4 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 645
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(b d-a e)^4 \left (a b+b^2 x\right )^5}{b^4}+\frac {4 e (b d-a e)^3 \left (a b+b^2 x\right )^6}{b^5}+\frac {6 e^2 (b d-a e)^2 \left (a b+b^2 x\right )^7}{b^6}+\frac {4 e^3 (b d-a e) \left (a b+b^2 x\right )^8}{b^7}+\frac {e^4 \left (a b+b^2 x\right )^9}{b^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}+\frac {4 e (b d-a e)^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 e^2 (b d-a e)^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^5}+\frac {4 e^3 (b d-a e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {e^4 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^5}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 319, normalized size = 1.46 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (252 a^5 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+210 a^4 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+120 a^3 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+10 a b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.14, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 360, normalized size = 1.64 \begin {gather*} \frac {1}{10} \, b^{5} e^{4} x^{10} + a^{5} d^{4} x + \frac {1}{9} \, {\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{4} + 4 \, a^{5} d^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 564, normalized size = 2.58 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{9} \, b^{5} d x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, b^{5} d^{2} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, b^{5} d^{3} x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, a b^{4} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a b^{4} d x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {30}{7} \, a b^{4} d^{2} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a b^{4} d^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{2} b^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {40}{7} \, a^{2} b^{3} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{2} b^{3} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, a^{3} b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a^{3} b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{3} b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, a^{4} b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{4} b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{4} b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a^{4} b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{5} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{5} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{5} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{5} d^{4} 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 [B] time = 0.05, size = 414, normalized size = 1.89 \begin {gather*} \frac {\left (126 e^{4} b^{5} x^{9}+700 x^{8} e^{4} a \,b^{4}+560 x^{8} d \,e^{3} b^{5}+1575 x^{7} e^{4} a^{2} b^{3}+3150 x^{7} d \,e^{3} a \,b^{4}+945 x^{7} d^{2} e^{2} b^{5}+1800 x^{6} e^{4} a^{3} b^{2}+7200 x^{6} d \,e^{3} a^{2} b^{3}+5400 x^{6} d^{2} e^{2} a \,b^{4}+720 x^{6} d^{3} e \,b^{5}+1050 x^{5} e^{4} a^{4} b +8400 x^{5} d \,e^{3} a^{3} b^{2}+12600 x^{5} d^{2} e^{2} a^{2} b^{3}+4200 x^{5} d^{3} e a \,b^{4}+210 x^{5} d^{4} b^{5}+252 x^{4} e^{4} a^{5}+5040 x^{4} d \,e^{3} a^{4} b +15120 x^{4} d^{2} e^{2} a^{3} b^{2}+10080 x^{4} d^{3} e \,a^{2} b^{3}+1260 x^{4} d^{4} a \,b^{4}+1260 x^{3} d \,e^{3} a^{5}+9450 x^{3} d^{2} e^{2} a^{4} b +12600 x^{3} d^{3} e \,a^{3} b^{2}+3150 x^{3} d^{4} a^{2} b^{3}+2520 x^{2} d^{2} e^{2} a^{5}+8400 x^{2} d^{3} e \,a^{4} b +4200 x^{2} d^{4} a^{3} b^{2}+2520 x \,d^{3} e \,a^{5}+3150 x \,d^{4} a^{4} b +1260 d^{4} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{1260 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.23, size = 588, normalized size = 2.68 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{4} x^{3}}{10 \, b^{2}} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{4} x - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3} e x}{3 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{2} e^{2} x}{b^{2}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d e^{3} x}{3 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{4} x}{6 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e^{3} x^{2}}{9 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{4} x^{2}}{90 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{4}}{6 \, b} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{3} e}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d^{2} e^{2}}{b^{3}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} d e^{3}}{3 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{4}}{6 \, b^{5}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{2} e^{2} x}{4 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d e^{3} x}{18 \, b^{3}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{4} x}{180 \, b^{4}} + \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{3} e}{7 \, b^{2}} - \frac {27 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d^{2} e^{2}}{28 \, b^{3}} + \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} d e^{3}}{126 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} e^{4}}{1260 \, b^{5}} \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 {\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,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 )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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