Optimal. Leaf size=344 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac {b^6 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{e^6 (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)} \]
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Rubi [A] time = 0.24, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} -\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{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)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac {b^6 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 21
Rule 43
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 {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^5} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^5} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac {b^5 (5 b d-6 a e)}{e^6}+\frac {b^6 x}{e^5}+\frac {(-b d+a e)^6}{e^6 (d+e x)^5}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^4}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^3}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^2}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {b^5 (5 b d-6 a e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^6 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {2 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^3}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {15 b^4 (b d-a e)^2 \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 = 0.15, size = 318, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 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 )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)+b^6 \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 )}{4 e^7 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.02, 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 = 0.43, size = 571, normalized size = 1.66 \begin {gather*} \frac {2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \, {\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \, {\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \, {\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \, {\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\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 [A] time = 0.23, size = 504, normalized size = 1.47 \begin {gather*} 15 \, {\left (b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (b^{6} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, b^{6} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 12 \, a b^{5} x e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-10\right )} + \frac {{\left (57 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 154 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 125 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 80 \, {\left (b^{6} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{3} b^{3} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 30 \, {\left (7 \, b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{2} b^{4} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{4} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 4 \, {\left (47 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 130 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 110 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 5 \, a^{4} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{4 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 670, normalized size = 1.95 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (2 b^{6} e^{6} x^{6}+60 a^{2} b^{4} e^{6} x^{4} \ln \left (e x +d \right )-120 a \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )+24 a \,b^{5} e^{6} x^{5}+60 b^{6} d^{2} e^{4} x^{4} \ln \left (e x +d \right )-12 b^{6} d \,e^{5} x^{5}+240 a^{2} b^{4} d \,e^{5} x^{3} \ln \left (e x +d \right )-480 a \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+96 a \,b^{5} d \,e^{5} x^{4}+240 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-68 b^{6} d^{2} e^{4} x^{4}-80 a^{3} b^{3} e^{6} x^{3}+360 a^{2} b^{4} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+240 a^{2} b^{4} d \,e^{5} x^{3}-720 a \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-96 a \,b^{5} d^{2} e^{4} x^{3}+360 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-32 b^{6} d^{3} e^{3} x^{3}-30 a^{4} b^{2} e^{6} x^{2}-120 a^{3} b^{3} d \,e^{5} x^{2}+240 a^{2} b^{4} d^{3} e^{3} x \ln \left (e x +d \right )+540 a^{2} b^{4} d^{2} e^{4} x^{2}-480 a \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-504 a \,b^{5} d^{3} e^{3} x^{2}+240 b^{6} d^{5} e x \ln \left (e x +d \right )+132 b^{6} d^{4} e^{2} x^{2}-8 a^{5} b \,e^{6} x -20 a^{4} b^{2} d \,e^{5} x -80 a^{3} b^{3} d^{2} e^{4} x +60 a^{2} b^{4} d^{4} e^{2} \ln \left (e x +d \right )+440 a^{2} b^{4} d^{3} e^{3} x -120 a \,b^{5} d^{5} e \ln \left (e x +d \right )-496 a \,b^{5} d^{4} e^{2} x +60 b^{6} d^{6} \ln \left (e x +d \right )+168 b^{6} d^{5} e x -a^{6} e^{6}-2 a^{5} b d \,e^{5}-5 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+125 a^{2} b^{4} d^{4} e^{2}-154 a \,b^{5} d^{5} e +57 b^{6} d^{6}\right )}{4 \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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