Optimal. Leaf size=421 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{4 e^7 (a+b x) (d+e x)^4}-\frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+5 b B d)}{e^6 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \]
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Rubi [A] time = 0.39, antiderivative size = 421, 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 x \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+5 b B d)}{e^6 (a+b x)}+\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)}{e^7 (a+b x) (d+e x)}-\frac {5 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)^2}+\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)}{3 e^7 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{4 e^7 (a+b x) (d+e x)^4}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \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)^5} \, 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)^5} \, 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^9 (-5 b B d+A b e+5 a B e)}{e^6}+\frac {b^{10} B x}{e^5}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^5}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^4}-\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)^3}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^2}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {b^4 (5 b B d-A b e-5 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\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}}{3 e^7 (a+b x) (d+e x)^3}-\frac {5 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)^2}+\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}}{e^7 (a+b x) (d+e x)}+\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} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 497, normalized size = 1.18 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5 e^5 (3 A e+B (d+4 e x))+5 a^4 b e^4 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-5 a b^4 e \left (A d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-B \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )-60 b^3 (d+e x)^4 (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)+b^5 \left (A e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-3 B \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )\right )}{12 e^7 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.42, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.46, size = 871, normalized size = 2.07 \begin {gather*} \frac {6 \, B b^{5} e^{6} x^{6} + 171 \, B b^{5} d^{6} - 3 \, A a^{5} e^{6} - 77 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 125 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 12 \, {\left (3 \, B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} - 12 \, {\left (17 \, B b^{5} d^{2} e^{4} - 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5}\right )} x^{4} - 24 \, {\left (4 \, B b^{5} d^{3} e^{3} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} - 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 6 \, {\left (66 \, B b^{5} d^{4} e^{2} - 42 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 90 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 4 \, {\left (126 \, B b^{5} d^{5} e - 62 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 110 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \, {\left (3 \, B b^{5} d^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + {\left (3 \, B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 4 \, {\left (3 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5}\right )} x^{3} + 6 \, {\left (3 \, B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (3 \, B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 870, normalized size = 2.07
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1163, normalized size = 2.76 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (6 B \,b^{5} e^{6} x^{6}+60 A a \,b^{4} e^{6} x^{4} \ln \left (e x +d \right )-60 A \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )+12 A \,b^{5} e^{6} x^{5}+120 B \,a^{2} b^{3} e^{6} x^{4} \ln \left (e x +d \right )-300 B a \,b^{4} d \,e^{5} x^{4} \ln \left (e x +d \right )+60 B a \,b^{4} e^{6} x^{5}+180 B \,b^{5} d^{2} e^{4} x^{4} \ln \left (e x +d \right )-36 B \,b^{5} d \,e^{5} x^{5}+240 A a \,b^{4} d \,e^{5} x^{3} \ln \left (e x +d \right )-240 A \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+48 A \,b^{5} d \,e^{5} x^{4}+480 B \,a^{2} b^{3} d \,e^{5} x^{3} \ln \left (e x +d \right )-1200 B a \,b^{4} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+240 B a \,b^{4} d \,e^{5} x^{4}+720 B \,b^{5} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-204 B \,b^{5} d^{2} e^{4} x^{4}-120 A \,a^{2} b^{3} e^{6} x^{3}+360 A a \,b^{4} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+240 A a \,b^{4} d \,e^{5} x^{3}-360 A \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-48 A \,b^{5} d^{2} e^{4} x^{3}-120 B \,a^{3} b^{2} e^{6} x^{3}+720 B \,a^{2} b^{3} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+480 B \,a^{2} b^{3} d \,e^{5} x^{3}-1800 B a \,b^{4} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-240 B a \,b^{4} d^{2} e^{4} x^{3}+1080 B \,b^{5} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-96 B \,b^{5} d^{3} e^{3} x^{3}-60 A \,a^{3} b^{2} e^{6} x^{2}-180 A \,a^{2} b^{3} d \,e^{5} x^{2}+240 A a \,b^{4} d^{3} e^{3} x \ln \left (e x +d \right )+540 A a \,b^{4} d^{2} e^{4} x^{2}-240 A \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-252 A \,b^{5} d^{3} e^{3} x^{2}-30 B \,a^{4} b \,e^{6} x^{2}-180 B \,a^{3} b^{2} d \,e^{5} x^{2}+480 B \,a^{2} b^{3} d^{3} e^{3} x \ln \left (e x +d \right )+1080 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-1200 B a \,b^{4} d^{4} e^{2} x \ln \left (e x +d \right )-1260 B a \,b^{4} d^{3} e^{3} x^{2}+720 B \,b^{5} d^{5} e x \ln \left (e x +d \right )+396 B \,b^{5} d^{4} e^{2} x^{2}-20 A \,a^{4} b \,e^{6} x -40 A \,a^{3} b^{2} d \,e^{5} x -120 A \,a^{2} b^{3} d^{2} e^{4} x +60 A a \,b^{4} d^{4} e^{2} \ln \left (e x +d \right )+440 A a \,b^{4} d^{3} e^{3} x -60 A \,b^{5} d^{5} e \ln \left (e x +d \right )-248 A \,b^{5} d^{4} e^{2} x -4 B \,a^{5} e^{6} x -20 B \,a^{4} b d \,e^{5} x -120 B \,a^{3} b^{2} d^{2} e^{4} x +120 B \,a^{2} b^{3} d^{4} e^{2} \ln \left (e x +d \right )+880 B \,a^{2} b^{3} d^{3} e^{3} x -300 B a \,b^{4} d^{5} e \ln \left (e x +d \right )-1240 B a \,b^{4} d^{4} e^{2} x +180 B \,b^{5} d^{6} \ln \left (e x +d \right )+504 B \,b^{5} d^{5} e x -3 A \,a^{5} e^{6}-5 A \,a^{4} b d \,e^{5}-10 A \,a^{3} b^{2} d^{2} e^{4}-30 A \,a^{2} b^{3} d^{3} e^{3}+125 A a \,b^{4} d^{4} e^{2}-77 A \,b^{5} d^{5} e -B \,a^{5} d \,e^{5}-5 B \,a^{4} b \,d^{2} e^{4}-30 B \,a^{3} b^{2} d^{3} e^{3}+250 B \,a^{2} b^{3} d^{4} e^{2}-385 B a \,b^{4} d^{5} e +171 B \,b^{5} d^{6}\right )}{12 \left (b x +a \right )^{5} \left (e x +d \right )^{4} 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 [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^5} \,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 (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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