Optimal. Leaf size=438 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \]
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Rubi [A] time = 0.30, antiderivative size = 438, 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 {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}+\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{13}} \, 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 {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{13}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{12}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^{11}}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^{10}}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^9}+\frac {b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^8}+\frac {b^{10} B}{e^6 (d+e x)^7}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{12 e^7 (a+b x) (d+e x)^{12}}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}-\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac {b^4 (6 b B d-A b e-5 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 465, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (42 a^5 e^5 (11 A e+B (d+12 e x))+42 a^4 b e^4 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+28 a^3 b^2 e^3 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+14 a^2 b^3 e^2 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )\right )+a b^4 e \left (7 A e \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+B \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )\right )}{5544 e^7 (a+b x) (d+e x)^{12}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.13, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 672, normalized size = 1.53 \begin {gather*} -\frac {924 \, B b^{5} e^{6} x^{6} + B b^{5} d^{6} + 462 \, A a^{5} e^{6} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 42 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 792 \, {\left (B b^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 495 \, {\left (B b^{5} d^{2} e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 220 \, {\left (B b^{5} d^{3} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 66 \, {\left (B b^{5} d^{4} e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 12 \, {\left (B b^{5} d^{5} e + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 42 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 917, normalized size = 2.09
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 687, normalized size = 1.57 \begin {gather*} -\frac {\left (924 B \,b^{5} e^{6} x^{6}+792 A \,b^{5} e^{6} x^{5}+3960 B a \,b^{4} e^{6} x^{5}+792 B \,b^{5} d \,e^{5} x^{5}+3465 A a \,b^{4} e^{6} x^{4}+495 A \,b^{5} d \,e^{5} x^{4}+6930 B \,a^{2} b^{3} e^{6} x^{4}+2475 B a \,b^{4} d \,e^{5} x^{4}+495 B \,b^{5} d^{2} e^{4} x^{4}+6160 A \,a^{2} b^{3} e^{6} x^{3}+1540 A a \,b^{4} d \,e^{5} x^{3}+220 A \,b^{5} d^{2} e^{4} x^{3}+6160 B \,a^{3} b^{2} e^{6} x^{3}+3080 B \,a^{2} b^{3} d \,e^{5} x^{3}+1100 B a \,b^{4} d^{2} e^{4} x^{3}+220 B \,b^{5} d^{3} e^{3} x^{3}+5544 A \,a^{3} b^{2} e^{6} x^{2}+1848 A \,a^{2} b^{3} d \,e^{5} x^{2}+462 A a \,b^{4} d^{2} e^{4} x^{2}+66 A \,b^{5} d^{3} e^{3} x^{2}+2772 B \,a^{4} b \,e^{6} x^{2}+1848 B \,a^{3} b^{2} d \,e^{5} x^{2}+924 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+330 B a \,b^{4} d^{3} e^{3} x^{2}+66 B \,b^{5} d^{4} e^{2} x^{2}+2520 A \,a^{4} b \,e^{6} x +1008 A \,a^{3} b^{2} d \,e^{5} x +336 A \,a^{2} b^{3} d^{2} e^{4} x +84 A a \,b^{4} d^{3} e^{3} x +12 A \,b^{5} d^{4} e^{2} x +504 B \,a^{5} e^{6} x +504 B \,a^{4} b d \,e^{5} x +336 B \,a^{3} b^{2} d^{2} e^{4} x +168 B \,a^{2} b^{3} d^{3} e^{3} x +60 B a \,b^{4} d^{4} e^{2} x +12 B \,b^{5} d^{5} e x +462 A \,a^{5} e^{6}+210 A \,a^{4} b d \,e^{5}+84 A \,a^{3} b^{2} d^{2} e^{4}+28 A \,a^{2} b^{3} d^{3} e^{3}+7 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +42 B \,a^{5} d \,e^{5}+42 B \,a^{4} b \,d^{2} e^{4}+28 B \,a^{3} b^{2} d^{3} e^{3}+14 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 \left (e x +d \right )^{12} \left (b x +a \right )^{5} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.63, size = 1489, normalized size = 3.40
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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