Optimal. Leaf size=302 \[ \frac{e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{20 e^3 (b d-a e)^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^6 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.238804, antiderivative size = 302, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 646
Rule 43
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*}
Mathematica [A] time = 0.147565, size = 313, normalized size = 1.04 \[ \frac{-a^2 b^4 e^2 \left (-540 d^2 e^2 x^2+80 d^3 e x+5 d^4+96 d e^3 x^3+68 e^4 x^4\right )-4 a^3 b^3 e^3 \left (-110 d^2 e x+5 d^3+126 d e^2 x^2+8 e^3 x^3\right )+a^4 b^2 e^4 \left (125 d^2-496 d e x+132 e^2 x^2\right )+14 a^5 b e^5 (12 e x-11 d)+57 a^6 e^6-2 a b^5 e \left (60 d^3 e^2 x^2-120 d^2 e^3 x^3+10 d^4 e x+d^5-48 d e^4 x^4+6 e^5 x^5\right )+60 e^4 (a+b x)^4 (b d-a e)^2 \log (a+b x)+b^6 \left (-\left (30 d^4 e^2 x^2+80 d^3 e^3 x^3+8 d^5 e x+d^6-24 d e^5 x^5-2 e^6 x^6\right )\right )}{4 b^7 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.205, size = 661, normalized size = 2.2 \begin{align*}{\frac{ \left ( -154\,{a}^{5}bd{e}^{5}+57\,{a}^{6}{e}^{6}-30\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+168\,x{a}^{5}b{e}^{6}-8\,x{b}^{6}{d}^{5}e-32\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-80\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+132\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-12\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-68\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-120\,\ln \left ( bx+a \right ){x}^{4}a{b}^{5}d{e}^{5}-2\,a{b}^{5}{d}^{5}e+540\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-120\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+440\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-80\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-20\,xa{b}^{5}{d}^{4}{e}^{2}+240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+60\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}{e}^{6}+60\,\ln \left ( bx+a \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}{e}^{6}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}{e}^{6}-{d}^{6}{b}^{6}+2\,{x}^{6}{b}^{6}{e}^{6}+240\,\ln \left ( bx+a \right ) x{a}^{5}b{e}^{6}-120\,\ln \left ( bx+a \right ){a}^{5}bd{e}^{5}+60\,\ln \left ( bx+a \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}-96\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-504\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-496\,x{a}^{4}{b}^{2}d{e}^{5}+96\,{x}^{4}a{b}^{5}d{e}^{5}+60\,\ln \left ( bx+a \right ){a}^{6}{e}^{6}-5\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+125\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-20\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}-480\,\ln \left ( bx+a \right ) x{a}^{4}{b}^{2}d{e}^{5}+240\,\ln \left ( bx+a \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-480\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}-720\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4} \right ) \left ( bx+a \right ) }{4\,{b}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.40594, size = 844, normalized size = 2.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.65025, size = 1158, normalized size = 3.83 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]