Optimal. Leaf size=262 \[ -\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 e (b d-a e)^3 (d+e x)^7}+\frac {b^2 (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 e (b d-a e)^4 (d+e x)^6} \]
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Rubi [A]
time = 0.12, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {784, 79, 47, 37}
\begin {gather*} \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{504 e (d+e x)^6 (b d-a e)^4}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{84 e (d+e x)^7 (b d-a e)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{24 e (d+e x)^8 (b d-a e)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 784
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)^{10}} \, 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)^{10}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac {\left ((2 b B d+A b e-3 a B e) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^9} \, dx}{3 b^4 e (b d-a e) \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac {\left ((2 b B d+A b e-3 a B e) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^8} \, dx}{12 b^3 e (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 e (b d-a e)^3 (d+e x)^7}+\frac {\left ((2 b B d+A b e-3 a B e) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{84 b^2 e (b d-a e)^3 \left (a b+b^2 x\right )}\\ &=-\frac {(B d-A e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 e (b d-a e)^3 (d+e x)^7}+\frac {b^2 (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 e (b d-a e)^4 (d+e x)^6}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 468, normalized size = 1.79 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (7 a^5 e^5 (8 A e+B (d+9 e x))+5 a^4 b e^4 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+2 a^2 b^3 e^2 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+a b^4 e \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )\right )}{504 e^7 (a+b x) (d+e x)^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(687\) vs.
\(2(210)=420\).
time = 0.81, size = 688, normalized size = 2.63
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} B \,x^{6}}{3 e}-\frac {b^{4} \left (A b e +5 B a e +2 B b d \right ) x^{5}}{4 e^{2}}-\frac {b^{3} \left (4 A a b \,e^{2}+A \,b^{2} d e +8 a^{2} B \,e^{2}+5 B a b d e +2 b^{2} B \,d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{2} \left (10 A \,a^{2} b \,e^{3}+4 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +10 B \,e^{3} a^{3}+8 B \,a^{2} b d \,e^{2}+5 B a \,b^{2} d^{2} e +2 B \,b^{3} d^{3}\right ) x^{3}}{6 e^{4}}-\frac {b \left (20 A \,a^{3} b \,e^{4}+10 A \,a^{2} b^{2} d \,e^{3}+4 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +10 B \,a^{4} e^{4}+10 B \,a^{3} b d \,e^{3}+8 B \,a^{2} b^{2} d^{2} e^{2}+5 B a \,b^{3} d^{3} e +2 B \,b^{4} d^{4}\right ) x^{2}}{14 e^{5}}-\frac {\left (35 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}+10 A \,a^{2} b^{3} d^{2} e^{3}+4 A a \,b^{4} d^{3} e^{2}+A \,b^{5} d^{4} e +7 B \,a^{5} e^{5}+10 B \,a^{4} b d \,e^{4}+10 B \,a^{3} b^{2} d^{2} e^{3}+8 B \,a^{2} b^{3} d^{3} e^{2}+5 B a \,b^{4} d^{4} e +2 b^{5} B \,d^{5}\right ) x}{56 e^{6}}-\frac {56 a^{5} A \,e^{6}+35 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+4 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +7 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+8 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +2 b^{5} B \,d^{6}}{504 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{9}}\) | \(596\) |
gosper | \(-\frac {\left (168 B \,b^{5} e^{6} x^{6}+126 A \,b^{5} e^{6} x^{5}+630 B a \,b^{4} e^{6} x^{5}+252 B \,b^{5} d \,e^{5} x^{5}+504 A a \,b^{4} e^{6} x^{4}+126 A \,b^{5} d \,e^{5} x^{4}+1008 B \,a^{2} b^{3} e^{6} x^{4}+630 B a \,b^{4} d \,e^{5} x^{4}+252 B \,b^{5} d^{2} e^{4} x^{4}+840 A \,a^{2} b^{3} e^{6} x^{3}+336 A a \,b^{4} d \,e^{5} x^{3}+84 A \,b^{5} d^{2} e^{4} x^{3}+840 B \,a^{3} b^{2} e^{6} x^{3}+672 B \,a^{2} b^{3} d \,e^{5} x^{3}+420 B a \,b^{4} d^{2} e^{4} x^{3}+168 B \,b^{5} d^{3} e^{3} x^{3}+720 A \,a^{3} b^{2} e^{6} x^{2}+360 A \,a^{2} b^{3} d \,e^{5} x^{2}+144 A a \,b^{4} d^{2} e^{4} x^{2}+36 A \,b^{5} d^{3} e^{3} x^{2}+360 B \,a^{4} b \,e^{6} x^{2}+360 B \,a^{3} b^{2} d \,e^{5} x^{2}+288 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+180 B a \,b^{4} d^{3} e^{3} x^{2}+72 B \,b^{5} d^{4} e^{2} x^{2}+315 A \,a^{4} b \,e^{6} x +180 A \,a^{3} b^{2} d \,e^{5} x +90 A \,a^{2} b^{3} d^{2} e^{4} x +36 A a \,b^{4} d^{3} e^{3} x +9 A \,b^{5} d^{4} e^{2} x +63 B \,a^{5} e^{6} x +90 B \,a^{4} b d \,e^{5} x +90 B \,a^{3} b^{2} d^{2} e^{4} x +72 B \,a^{2} b^{3} d^{3} e^{3} x +45 B a \,b^{4} d^{4} e^{2} x +18 B \,b^{5} d^{5} e x +56 a^{5} A \,e^{6}+35 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+4 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +7 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+8 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +2 b^{5} B \,d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{7} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(688\) |
default | \(-\frac {\left (168 B \,b^{5} e^{6} x^{6}+126 A \,b^{5} e^{6} x^{5}+630 B a \,b^{4} e^{6} x^{5}+252 B \,b^{5} d \,e^{5} x^{5}+504 A a \,b^{4} e^{6} x^{4}+126 A \,b^{5} d \,e^{5} x^{4}+1008 B \,a^{2} b^{3} e^{6} x^{4}+630 B a \,b^{4} d \,e^{5} x^{4}+252 B \,b^{5} d^{2} e^{4} x^{4}+840 A \,a^{2} b^{3} e^{6} x^{3}+336 A a \,b^{4} d \,e^{5} x^{3}+84 A \,b^{5} d^{2} e^{4} x^{3}+840 B \,a^{3} b^{2} e^{6} x^{3}+672 B \,a^{2} b^{3} d \,e^{5} x^{3}+420 B a \,b^{4} d^{2} e^{4} x^{3}+168 B \,b^{5} d^{3} e^{3} x^{3}+720 A \,a^{3} b^{2} e^{6} x^{2}+360 A \,a^{2} b^{3} d \,e^{5} x^{2}+144 A a \,b^{4} d^{2} e^{4} x^{2}+36 A \,b^{5} d^{3} e^{3} x^{2}+360 B \,a^{4} b \,e^{6} x^{2}+360 B \,a^{3} b^{2} d \,e^{5} x^{2}+288 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+180 B a \,b^{4} d^{3} e^{3} x^{2}+72 B \,b^{5} d^{4} e^{2} x^{2}+315 A \,a^{4} b \,e^{6} x +180 A \,a^{3} b^{2} d \,e^{5} x +90 A \,a^{2} b^{3} d^{2} e^{4} x +36 A a \,b^{4} d^{3} e^{3} x +9 A \,b^{5} d^{4} e^{2} x +63 B \,a^{5} e^{6} x +90 B \,a^{4} b d \,e^{5} x +90 B \,a^{3} b^{2} d^{2} e^{4} x +72 B \,a^{2} b^{3} d^{3} e^{3} x +45 B a \,b^{4} d^{4} e^{2} x +18 B \,b^{5} d^{5} e x +56 a^{5} A \,e^{6}+35 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+4 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +7 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+8 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +2 b^{5} B \,d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{7} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\) | \(688\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 609 vs.
\(2 (221) = 442\).
time = 1.51, size = 609, normalized size = 2.32 \begin {gather*} -\frac {2 \, B b^{5} d^{6} + {\left (168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )} e^{6} + {\left (252 \, B b^{5} d x^{5} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d x^{4} + 336 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d x^{3} + 360 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d x^{2} + 90 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d x + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d\right )} e^{5} + 2 \, {\left (126 \, B b^{5} d^{2} x^{4} + 42 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} x^{3} + 72 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} x^{2} + 45 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} x + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2}\right )} e^{4} + 2 \, {\left (84 \, B b^{5} d^{3} x^{3} + 18 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} x^{2} + 18 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} x + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3}\right )} e^{3} + {\left (72 \, B b^{5} d^{4} x^{2} + 9 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} x + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4}\right )} e^{2} + {\left (18 \, B b^{5} d^{5} x + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5}\right )} e}{504 \, {\left (x^{9} e^{16} + 9 \, d x^{8} e^{15} + 36 \, d^{2} x^{7} e^{14} + 84 \, d^{3} x^{6} e^{13} + 126 \, d^{4} x^{5} e^{12} + 126 \, d^{5} x^{4} e^{11} + 84 \, d^{6} x^{3} e^{10} + 36 \, d^{7} x^{2} e^{9} + 9 \, d^{8} x e^{8} + d^{9} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 918 vs.
\(2 (221) = 442\).
time = 1.92, size = 918, normalized size = 3.50 \begin {gather*} -\frac {{\left (168 \, B b^{5} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 252 \, B b^{5} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 252 \, B b^{5} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 168 \, B b^{5} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 72 \, B b^{5} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 18 \, B b^{5} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + 2 \, B b^{5} d^{6} \mathrm {sgn}\left (b x + a\right ) + 630 \, B a b^{4} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 126 \, A b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 630 \, B a b^{4} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, A b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 420 \, B a b^{4} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, A b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 180 \, B a b^{4} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, A b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, B a b^{4} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, A b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a b^{4} d^{5} e \mathrm {sgn}\left (b x + a\right ) + A b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 1008 \, B a^{2} b^{3} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 504 \, A a b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 672 \, B a^{2} b^{3} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 336 \, A a b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 288 \, B a^{2} b^{3} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 144 \, A a b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 72 \, B a^{2} b^{3} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, A a b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 8 \, B a^{2} b^{3} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, A a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 840 \, B a^{3} b^{2} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{2} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 360 \, B a^{3} b^{2} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 360 \, A a^{2} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 90 \, B a^{3} b^{2} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 90 \, A a^{2} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, B a^{3} b^{2} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, A a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 360 \, B a^{4} b x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 720 \, A a^{3} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 90 \, B a^{4} b d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 180 \, A a^{3} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 10 \, B a^{4} b d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, A a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 63 \, B a^{5} x e^{6} \mathrm {sgn}\left (b x + a\right ) + 315 \, A a^{4} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, B a^{5} d e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, A a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 56 \, A a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{504 \, {\left (x e + d\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.64, size = 1489, normalized size = 5.68 \begin {gather*} -\frac {\left (\frac {10\,B\,a^2\,b^3\,e^2-20\,B\,a\,b^4\,d\,e+5\,A\,a\,b^4\,e^2+10\,B\,b^5\,d^2-4\,A\,b^5\,d\,e}{5\,e^7}-\frac {d\,\left (\frac {b^4\,\left (A\,b\,e+5\,B\,a\,e-4\,B\,b\,d\right )}{5\,e^6}-\frac {B\,b^5\,d}{5\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {A\,b^5\,e-5\,B\,b^5\,d+5\,B\,a\,b^4\,e}{4\,e^7}-\frac {B\,b^5\,d}{4\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {\left (\frac {A\,a^5}{9\,e}-\frac {d\,\left (\frac {B\,a^5+5\,A\,b\,a^4}{9\,e}+\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {A\,b^5+5\,B\,a\,b^4}{9\,e}-\frac {B\,b^5\,d}{9\,e^2}\right )}{e}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )}{9\,e}\right )}{e}+\frac {10\,a^2\,b^2\,\left (A\,b+B\,a\right )}{9\,e}\right )}{e}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )}{9\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}-\frac {\left (\frac {10\,B\,a^3\,b^2\,e^3-30\,B\,a^2\,b^3\,d\,e^2+10\,A\,a^2\,b^3\,e^3+30\,B\,a\,b^4\,d^2\,e-15\,A\,a\,b^4\,d\,e^2-10\,B\,b^5\,d^3+6\,A\,b^5\,d^2\,e}{6\,e^7}-\frac {d\,\left (\frac {10\,B\,a^2\,b^3\,e^3-15\,B\,a\,b^4\,d\,e^2+5\,A\,a\,b^4\,e^3+6\,B\,b^5\,d^2\,e-3\,A\,b^5\,d\,e^2}{6\,e^7}-\frac {d\,\left (\frac {b^4\,\left (A\,b\,e+5\,B\,a\,e-3\,B\,b\,d\right )}{6\,e^5}-\frac {B\,b^5\,d}{6\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {B\,a^5\,e^5-5\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5+10\,B\,a^3\,b^2\,d^2\,e^3-10\,A\,a^3\,b^2\,d\,e^4-10\,B\,a^2\,b^3\,d^3\,e^2+10\,A\,a^2\,b^3\,d^2\,e^3+5\,B\,a\,b^4\,d^4\,e-5\,A\,a\,b^4\,d^3\,e^2-B\,b^5\,d^5+A\,b^5\,d^4\,e}{8\,e^7}-\frac {d\,\left (\frac {5\,B\,a^4\,b\,e^5-10\,B\,a^3\,b^2\,d\,e^4+10\,A\,a^3\,b^2\,e^5+10\,B\,a^2\,b^3\,d^2\,e^3-10\,A\,a^2\,b^3\,d\,e^4-5\,B\,a\,b^4\,d^3\,e^2+5\,A\,a\,b^4\,d^2\,e^3+B\,b^5\,d^4\,e-A\,b^5\,d^3\,e^2}{8\,e^7}-\frac {d\,\left (\frac {10\,B\,a^3\,b^2\,e^5-10\,B\,a^2\,b^3\,d\,e^4+10\,A\,a^2\,b^3\,e^5+5\,B\,a\,b^4\,d^2\,e^3-5\,A\,a\,b^4\,d\,e^4-B\,b^5\,d^3\,e^2+A\,b^5\,d^2\,e^3}{8\,e^7}-\frac {d\,\left (\frac {10\,B\,a^2\,b^3\,e^5-5\,B\,a\,b^4\,d\,e^4+5\,A\,a\,b^4\,e^5+B\,b^5\,d^2\,e^3-A\,b^5\,d\,e^4}{8\,e^7}-\frac {d\,\left (\frac {b^4\,\left (A\,b\,e+5\,B\,a\,e-B\,b\,d\right )}{8\,e^3}-\frac {B\,b^5\,d}{8\,e^3}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {5\,B\,a^4\,b\,e^4-20\,B\,a^3\,b^2\,d\,e^3+10\,A\,a^3\,b^2\,e^4+30\,B\,a^2\,b^3\,d^2\,e^2-20\,A\,a^2\,b^3\,d\,e^3-20\,B\,a\,b^4\,d^3\,e+15\,A\,a\,b^4\,d^2\,e^2+5\,B\,b^5\,d^4-4\,A\,b^5\,d^3\,e}{7\,e^7}-\frac {d\,\left (\frac {10\,B\,a^3\,b^2\,e^4-20\,B\,a^2\,b^3\,d\,e^3+10\,A\,a^2\,b^3\,e^4+15\,B\,a\,b^4\,d^2\,e^2-10\,A\,a\,b^4\,d\,e^3-4\,B\,b^5\,d^3\,e+3\,A\,b^5\,d^2\,e^2}{7\,e^7}-\frac {d\,\left (\frac {10\,B\,a^2\,b^3\,e^4-10\,B\,a\,b^4\,d\,e^3+5\,A\,a\,b^4\,e^4+3\,B\,b^5\,d^2\,e^2-2\,A\,b^5\,d\,e^3}{7\,e^7}-\frac {d\,\left (\frac {b^4\,\left (A\,b\,e+5\,B\,a\,e-2\,B\,b\,d\right )}{7\,e^4}-\frac {B\,b^5\,d}{7\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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