Optimal. Leaf size=302 \[ \frac {15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {20 e^3 (b d-a e)^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^6 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.24, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac {e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {20 e^3 (b d-a e)^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^6 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {(d+e x)^6}{\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 {(d+e x)^6}{\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^5 (6 b d-5 a e)}{b^{11}}+\frac {e^6 x}{b^{10}}+\frac {(b d-a e)^6}{b^{11} (a+b x)^5}+\frac {6 e (b d-a e)^5}{b^{11} (a+b x)^4}+\frac {15 e^2 (b d-a e)^4}{b^{11} (a+b x)^3}+\frac {20 e^3 (b d-a e)^3}{b^{11} (a+b x)^2}+\frac {15 e^4 (b d-a e)^2}{b^{11} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {20 e^3 (b d-a e)^3}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^5 (6 b d-5 a e) x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^6 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^4 (b d-a e)^2 (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 = 0.15, size = 313, normalized size = 1.04 \[ \frac {57 a^6 e^6+14 a^5 b e^5 (12 e x-11 d)+a^4 b^2 e^4 \left (125 d^2-496 d e x+132 e^2 x^2\right )-4 a^3 b^3 e^3 \left (5 d^3-110 d^2 e x+126 d e^2 x^2+8 e^3 x^3\right )-a^2 b^4 e^2 \left (5 d^4+80 d^3 e x-540 d^2 e^2 x^2+96 d e^3 x^3+68 e^4 x^4\right )-2 a b^5 e \left (d^5+10 d^4 e x+60 d^3 e^2 x^2-120 d^2 e^3 x^3-48 d e^4 x^4+6 e^5 x^5\right )+60 e^4 (a+b x)^4 (b d-a e)^2 \log (a+b x)-\left (b^6 \left (d^6+8 d^5 e x+30 d^4 e^2 x^2+80 d^3 e^3 x^3-24 d e^5 x^5-2 e^6 x^6\right )\right )}{4 b^7 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 572, normalized size = 1.89 \[ \frac {2 \, b^{6} e^{6} x^{6} - b^{6} d^{6} - 2 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 125 \, a^{4} b^{2} d^{2} e^{4} - 154 \, a^{5} b d e^{5} + 57 \, a^{6} e^{6} + 12 \, {\left (2 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 4 \, {\left (24 \, a b^{5} d e^{5} - 17 \, a^{2} b^{4} e^{6}\right )} x^{4} - 16 \, {\left (5 \, b^{6} d^{3} e^{3} - 15 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{3} - 6 \, {\left (5 \, b^{6} d^{4} e^{2} + 20 \, a b^{5} d^{3} e^{3} - 90 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} - 22 \, a^{4} b^{2} e^{6}\right )} x^{2} - 4 \, {\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 124 \, a^{4} b^{2} d e^{5} - 42 \, a^{5} b e^{6}\right )} x + 60 \, {\left (a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} + a^{6} e^{6} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (a b^{5} d^{2} e^{4} - 2 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (a^{2} b^{4} d^{2} e^{4} - 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (a^{3} b^{3} d^{2} e^{4} - 2 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{4 \, {\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 661, normalized size = 2.19 \[ \frac {\left (2 b^{6} e^{6} x^{6}+60 a^{2} b^{4} e^{6} x^{4} \ln \left (b x +a \right )-120 a \,b^{5} d \,e^{5} x^{4} \ln \left (b x +a \right )-12 a \,b^{5} e^{6} x^{5}+60 b^{6} d^{2} e^{4} x^{4} \ln \left (b x +a \right )+24 b^{6} d \,e^{5} x^{5}+240 a^{3} b^{3} e^{6} x^{3} \ln \left (b x +a \right )-480 a^{2} b^{4} d \,e^{5} x^{3} \ln \left (b x +a \right )-68 a^{2} b^{4} e^{6} x^{4}+240 a \,b^{5} d^{2} e^{4} x^{3} \ln \left (b x +a \right )+96 a \,b^{5} d \,e^{5} x^{4}+360 a^{4} b^{2} e^{6} x^{2} \ln \left (b x +a \right )-720 a^{3} b^{3} d \,e^{5} x^{2} \ln \left (b x +a \right )-32 a^{3} b^{3} e^{6} x^{3}+360 a^{2} b^{4} d^{2} e^{4} x^{2} \ln \left (b x +a \right )-96 a^{2} b^{4} d \,e^{5} x^{3}+240 a \,b^{5} d^{2} e^{4} x^{3}-80 b^{6} d^{3} e^{3} x^{3}+240 a^{5} b \,e^{6} x \ln \left (b x +a \right )-480 a^{4} b^{2} d \,e^{5} x \ln \left (b x +a \right )+132 a^{4} b^{2} e^{6} x^{2}+240 a^{3} b^{3} d^{2} e^{4} x \ln \left (b x +a \right )-504 a^{3} b^{3} d \,e^{5} x^{2}+540 a^{2} b^{4} d^{2} e^{4} x^{2}-120 a \,b^{5} d^{3} e^{3} x^{2}-30 b^{6} d^{4} e^{2} x^{2}+60 a^{6} e^{6} \ln \left (b x +a \right )-120 a^{5} b d \,e^{5} \ln \left (b x +a \right )+168 a^{5} b \,e^{6} x +60 a^{4} b^{2} d^{2} e^{4} \ln \left (b x +a \right )-496 a^{4} b^{2} d \,e^{5} x +440 a^{3} b^{3} d^{2} e^{4} x -80 a^{2} b^{4} d^{3} e^{3} x -20 a \,b^{5} d^{4} e^{2} x -8 b^{6} d^{5} e x +57 a^{6} e^{6}-154 a^{5} b d \,e^{5}+125 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-5 a^{2} b^{4} d^{4} e^{2}-2 a \,b^{5} d^{5} e -b^{6} d^{6}\right ) \left (b x +a \right )}{4 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.57, size = 576, normalized size = 1.91 \[ \frac {1}{4} \, e^{6} {\left (\frac {2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6}}{b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}} + \frac {60 \, a^{2} \log \left (b x + a\right )}{b^{7}}\right )} + \frac {1}{2} \, d e^{5} {\left (\frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}} - \frac {60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac {5}{4} \, d^{2} e^{4} {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} - \frac {5}{3} \, d^{3} e^{3} {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {1}{2} \, d^{5} e {\left (\frac {4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {3 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {5}{4} \, d^{4} e^{2} {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {d^{6}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^6}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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