Optimal. Leaf size=259 \[ \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (-4 a B e+A b e+3 b B d)}{9 b^5}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{8 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{7 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^3}{6 b^5}+\frac {B e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \]
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Rubi [A] time = 0.46, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (-4 a B e+A b e+3 b B d)}{9 b^5}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{8 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{7 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^3}{6 b^5}+\frac {B e^3 \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 77
Rule 770
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^3 \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 (a b+b^2 x\right )^5 (A+B x) (d+e x)^3 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(A b-a B) (b d-a e)^3 \left (a b+b^2 x\right )^5}{b^4}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) \left (a b+b^2 x\right )^6}{b^5}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) \left (a b+b^2 x\right )^7}{b^6}+\frac {e^2 (3 b B d+A b e-4 a B e) \left (a b+b^2 x\right )^8}{b^7}+\frac {B e^3 \left (a b+b^2 x\right )^9}{b^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e)^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac {B e^3 (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 = 0.20, size = 478, normalized size = 1.85 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (126 a^5 \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+210 a^4 b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+60 a^3 b^2 x^2 \left (7 A \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )+90 a^2 b^3 x^3 \left (2 A \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )+5 a b^4 x^4 \left (9 A \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+5 B x \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )+b^5 x^5 \left (5 A \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+3 B x \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right )\right )}{2520 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 7.16, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^3 \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 = 0.44, size = 532, normalized size = 2.05 \begin {gather*} \frac {1}{10} \, B b^{5} e^{3} x^{10} + A a^{5} d^{3} x + \frac {1}{9} \, {\left (3 \, B b^{5} d e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (3 \, B b^{5} d^{2} e + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{2} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{3} + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{2} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e + 15 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{2} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{5} e^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} + 15 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{5} d e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{5} d^{2} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 934, normalized size = 3.61
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 676, normalized size = 2.61 \begin {gather*} \frac {\left (252 B \,e^{3} b^{5} x^{9}+280 x^{8} A \,b^{5} e^{3}+1400 x^{8} B \,e^{3} a \,b^{4}+840 x^{8} B \,b^{5} d \,e^{2}+1575 x^{7} A a \,b^{4} e^{3}+945 x^{7} A \,b^{5} d \,e^{2}+3150 x^{7} B \,e^{3} a^{2} b^{3}+4725 x^{7} B a \,b^{4} d \,e^{2}+945 x^{7} B \,b^{5} d^{2} e +3600 x^{6} A \,a^{2} b^{3} e^{3}+5400 x^{6} A a \,b^{4} d \,e^{2}+1080 x^{6} A \,b^{5} d^{2} e +3600 x^{6} B \,e^{3} a^{3} b^{2}+10800 x^{6} B \,a^{2} b^{3} d \,e^{2}+5400 x^{6} B a \,b^{4} d^{2} e +360 x^{6} B \,b^{5} d^{3}+4200 x^{5} A \,a^{3} b^{2} e^{3}+12600 x^{5} A \,a^{2} b^{3} d \,e^{2}+6300 x^{5} A a \,b^{4} d^{2} e +420 x^{5} A \,d^{3} b^{5}+2100 x^{5} B \,e^{3} a^{4} b +12600 x^{5} B \,a^{3} b^{2} d \,e^{2}+12600 x^{5} B \,a^{2} b^{3} d^{2} e +2100 x^{5} B a \,b^{4} d^{3}+2520 x^{4} A \,a^{4} b \,e^{3}+15120 x^{4} A \,a^{3} b^{2} d \,e^{2}+15120 x^{4} A \,a^{2} b^{3} d^{2} e +2520 x^{4} A \,d^{3} a \,b^{4}+504 x^{4} B \,e^{3} a^{5}+7560 x^{4} B \,a^{4} b d \,e^{2}+15120 x^{4} B \,a^{3} b^{2} d^{2} e +5040 x^{4} B \,a^{2} b^{3} d^{3}+630 x^{3} A \,a^{5} e^{3}+9450 x^{3} A \,a^{4} b d \,e^{2}+18900 x^{3} A \,a^{3} b^{2} d^{2} e +6300 x^{3} A \,d^{3} a^{2} b^{3}+1890 x^{3} B \,a^{5} d \,e^{2}+9450 x^{3} B \,a^{4} b \,d^{2} e +6300 x^{3} B \,a^{3} b^{2} d^{3}+2520 x^{2} A \,a^{5} d \,e^{2}+12600 x^{2} A \,a^{4} b \,d^{2} e +8400 x^{2} A \,d^{3} a^{3} b^{2}+2520 x^{2} B \,a^{5} d^{2} e +4200 x^{2} B \,a^{4} b \,d^{3}+3780 x A \,a^{5} d^{2} e +6300 x A \,d^{3} a^{4} b +1260 x B \,a^{5} d^{3}+2520 A \,d^{3} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{2520 \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 = 0.54, size = 698, normalized size = 2.69 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B e^{3} x^{3}}{10 \, b^{2}} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{4} e^{3} x}{6 \, b^{4}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a e^{3} x^{2}}{90 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d^{3}}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{5} e^{3}}{6 \, b^{5}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} e^{3} x}{180 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{3} e^{3}}{1260 \, b^{5}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x}{6 \, b^{3}} + \frac {{\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{6 \, b} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, b^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4}}{6 \, b^{4}} + \frac {{\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{6 \, b^{2}} - \frac {11 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x}{72 \, b^{3}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x}{8 \, b^{2}} + \frac {83 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2}}{504 \, b^{4}} - \frac {27 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a}{56 \, b^{3}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b^{2}} \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 (A+B\,x\right )\,{\left (d+e\,x\right )}^3\,{\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 (A + B x\right ) \left (d + e x\right )^{3} \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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