Optimal. Leaf size=373 \[ \frac {5 e^3 (a+b x) (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 e^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{3 b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 x (a+b x) (-5 a B e+A b e+5 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^5 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.43, antiderivative size = 373, 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^4 x (a+b x) (-5 a B e+A b e+5 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 e^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^3 (a+b x) (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{3 b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^5 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^5}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {e^4 (5 b B d+A b e-5 a B e)}{b^{11}}+\frac {B e^5 x}{b^{10}}+\frac {(A b-a B) (b d-a e)^5}{b^{11} (a+b x)^5}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^{11} (a+b x)^4}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{b^{11} (a+b x)^3}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e)}{b^{11} (a+b x)^2}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e)}{b^{11} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{3 b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (5 b B d+A b e-5 a B e) x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^5 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 513, normalized size = 1.38 \begin {gather*} \frac {-A b \left (77 a^5 e^5+a^4 b e^4 (248 e x-125 d)+2 a^3 b^2 e^3 \left (15 d^2-220 d e x+126 e^2 x^2\right )+2 a^2 b^3 e^2 \left (5 d^3+60 d^2 e x-270 d e^2 x^2+24 e^3 x^3\right )+a b^4 e \left (5 d^4+40 d^3 e x+180 d^2 e^2 x^2-240 d e^3 x^3-48 e^4 x^4\right )+b^5 \left (3 d^5+20 d^4 e x+60 d^3 e^2 x^2+120 d^2 e^3 x^3-12 e^5 x^5\right )\right )+B \left (171 a^6 e^5+7 a^5 b e^4 (72 e x-55 d)+2 a^4 b^2 e^3 \left (125 d^2-620 d e x+198 e^2 x^2\right )-2 a^3 b^3 e^2 \left (15 d^3-440 d^2 e x+630 d e^2 x^2+48 e^3 x^3\right )-a^2 b^4 e \left (5 d^4+120 d^3 e x-1080 d^2 e^2 x^2+240 d e^3 x^3+204 e^4 x^4\right )-a b^5 \left (d^5+20 d^4 e x+180 d^3 e^2 x^2-480 d^2 e^3 x^3-240 d e^4 x^4+36 e^5 x^5\right )+2 b^6 x \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )\right )+60 e^3 (a+b x)^4 (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{12 b^7 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 48.39, size = 14568, normalized size = 39.06 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 914, normalized size = 2.45 \begin {gather*} \frac {6 \, B b^{6} e^{5} x^{6} - {\left (B a b^{5} + 3 \, A b^{6}\right )} d^{5} - 5 \, {\left (B a^{2} b^{4} + A a b^{5}\right )} d^{4} e - 10 \, {\left (3 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} d^{3} e^{2} + 10 \, {\left (25 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} d^{2} e^{3} - 5 \, {\left (77 \, B a^{5} b - 25 \, A a^{4} b^{2}\right )} d e^{4} + {\left (171 \, B a^{6} - 77 \, A a^{5} b\right )} e^{5} + 12 \, {\left (5 \, B b^{6} d e^{4} - {\left (3 \, B a b^{5} - A b^{6}\right )} e^{5}\right )} x^{5} + 12 \, {\left (20 \, B a b^{5} d e^{4} - {\left (17 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} e^{5}\right )} x^{4} - 24 \, {\left (5 \, B b^{6} d^{3} e^{2} - 5 \, {\left (4 \, B a b^{5} - A b^{6}\right )} d^{2} e^{3} + 10 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d e^{4} + 2 \, {\left (2 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} e^{5}\right )} x^{3} - 6 \, {\left (5 \, B b^{6} d^{4} e + 10 \, {\left (3 \, B a b^{5} + A b^{6}\right )} d^{3} e^{2} - 30 \, {\left (6 \, B a^{2} b^{4} - A a b^{5}\right )} d^{2} e^{3} + 30 \, {\left (7 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d e^{4} - 6 \, {\left (11 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} - 4 \, {\left (B b^{6} d^{5} + 5 \, {\left (B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{4} + A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (22 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 10 \, {\left (31 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} d e^{4} - 2 \, {\left (63 \, B a^{5} b - 31 \, A a^{4} b^{2}\right )} e^{5}\right )} x + 60 \, {\left (2 \, B a^{4} b^{2} d^{2} e^{3} - {\left (5 \, B a^{5} b - A a^{4} b^{2}\right )} d e^{4} + {\left (3 \, B a^{6} - A a^{5} b\right )} e^{5} + {\left (2 \, B b^{6} d^{2} e^{3} - {\left (5 \, B a b^{5} - A b^{6}\right )} d e^{4} + {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} e^{5}\right )} x^{4} + 4 \, {\left (2 \, B a b^{5} d^{2} e^{3} - {\left (5 \, B a^{2} b^{4} - A a b^{5}\right )} d e^{4} + {\left (3 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 6 \, {\left (2 \, B a^{2} b^{4} d^{2} e^{3} - {\left (5 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} d e^{4} + {\left (3 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 4 \, {\left (2 \, B a^{3} b^{3} d^{2} e^{3} - {\left (5 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} d e^{4} + {\left (3 \, B a^{5} b - A a^{4} b^{2}\right )} e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1153, normalized size = 3.09 \begin {gather*} -\frac {\left (-6 B \,b^{6} e^{5} x^{6}+60 A a \,b^{5} e^{5} x^{4} \ln \left (b x +a \right )-60 A \,b^{6} d \,e^{4} x^{4} \ln \left (b x +a \right )-12 A \,b^{6} e^{5} x^{5}-180 B \,a^{2} b^{4} e^{5} x^{4} \ln \left (b x +a \right )+300 B a \,b^{5} d \,e^{4} x^{4} \ln \left (b x +a \right )+36 B a \,b^{5} e^{5} x^{5}-120 B \,b^{6} d^{2} e^{3} x^{4} \ln \left (b x +a \right )-60 B \,b^{6} d \,e^{4} x^{5}+240 A \,a^{2} b^{4} e^{5} x^{3} \ln \left (b x +a \right )-240 A a \,b^{5} d \,e^{4} x^{3} \ln \left (b x +a \right )-48 A a \,b^{5} e^{5} x^{4}-720 B \,a^{3} b^{3} e^{5} x^{3} \ln \left (b x +a \right )+1200 B \,a^{2} b^{4} d \,e^{4} x^{3} \ln \left (b x +a \right )+204 B \,a^{2} b^{4} e^{5} x^{4}-480 B a \,b^{5} d^{2} e^{3} x^{3} \ln \left (b x +a \right )-240 B a \,b^{5} d \,e^{4} x^{4}+360 A \,a^{3} b^{3} e^{5} x^{2} \ln \left (b x +a \right )-360 A \,a^{2} b^{4} d \,e^{4} x^{2} \ln \left (b x +a \right )+48 A \,a^{2} b^{4} e^{5} x^{3}-240 A a \,b^{5} d \,e^{4} x^{3}+120 A \,b^{6} d^{2} e^{3} x^{3}-1080 B \,a^{4} b^{2} e^{5} x^{2} \ln \left (b x +a \right )+1800 B \,a^{3} b^{3} d \,e^{4} x^{2} \ln \left (b x +a \right )+96 B \,a^{3} b^{3} e^{5} x^{3}-720 B \,a^{2} b^{4} d^{2} e^{3} x^{2} \ln \left (b x +a \right )+240 B \,a^{2} b^{4} d \,e^{4} x^{3}-480 B a \,b^{5} d^{2} e^{3} x^{3}+120 B \,b^{6} d^{3} e^{2} x^{3}+240 A \,a^{4} b^{2} e^{5} x \ln \left (b x +a \right )-240 A \,a^{3} b^{3} d \,e^{4} x \ln \left (b x +a \right )+252 A \,a^{3} b^{3} e^{5} x^{2}-540 A \,a^{2} b^{4} d \,e^{4} x^{2}+180 A a \,b^{5} d^{2} e^{3} x^{2}+60 A \,b^{6} d^{3} e^{2} x^{2}-720 B \,a^{5} b \,e^{5} x \ln \left (b x +a \right )+1200 B \,a^{4} b^{2} d \,e^{4} x \ln \left (b x +a \right )-396 B \,a^{4} b^{2} e^{5} x^{2}-480 B \,a^{3} b^{3} d^{2} e^{3} x \ln \left (b x +a \right )+1260 B \,a^{3} b^{3} d \,e^{4} x^{2}-1080 B \,a^{2} b^{4} d^{2} e^{3} x^{2}+180 B a \,b^{5} d^{3} e^{2} x^{2}+30 B \,b^{6} d^{4} e \,x^{2}+60 A \,a^{5} b \,e^{5} \ln \left (b x +a \right )-60 A \,a^{4} b^{2} d \,e^{4} \ln \left (b x +a \right )+248 A \,a^{4} b^{2} e^{5} x -440 A \,a^{3} b^{3} d \,e^{4} x +120 A \,a^{2} b^{4} d^{2} e^{3} x +40 A a \,b^{5} d^{3} e^{2} x +20 A \,b^{6} d^{4} e x -180 B \,a^{6} e^{5} \ln \left (b x +a \right )+300 B \,a^{5} b d \,e^{4} \ln \left (b x +a \right )-504 B \,a^{5} b \,e^{5} x -120 B \,a^{4} b^{2} d^{2} e^{3} \ln \left (b x +a \right )+1240 B \,a^{4} b^{2} d \,e^{4} x -880 B \,a^{3} b^{3} d^{2} e^{3} x +120 B \,a^{2} b^{4} d^{3} e^{2} x +20 B a \,b^{5} d^{4} e x +4 B \,b^{6} d^{5} x +77 A \,a^{5} b \,e^{5}-125 A \,a^{4} b^{2} d \,e^{4}+30 A \,a^{3} b^{3} d^{2} e^{3}+10 A \,a^{2} b^{4} d^{3} e^{2}+5 A a \,b^{5} d^{4} e +3 A \,b^{6} d^{5}-171 B \,a^{6} e^{5}+385 B \,a^{5} b d \,e^{4}-250 B \,a^{4} b^{2} d^{2} e^{3}+30 B \,a^{3} b^{3} d^{3} e^{2}+5 B \,a^{2} b^{4} d^{4} e +B a \,b^{5} d^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 1010, normalized size = 2.71
result too large to display
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 \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^5}{{\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 \frac {\left (A + B x\right ) \left (d + e x\right )^{5}}{\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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