Optimal. Leaf size=435 \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x) (d+e x)^9}+\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^{10}}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.294193, antiderivative size = 435, normalized size of antiderivative = 1., 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} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x) (d+e x)^9}+\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^{10}}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 77
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)^{14}} \, 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)^{14}} \, 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^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{14}}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{13}}-\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)^{12}}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^{11}}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^{10}}+\frac{b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^9}+\frac{b^{10} B}{e^6 (d+e x)^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{(b d-a e)^5 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\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}}{12 e^7 (a+b x) (d+e x)^{12}}-\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}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{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)^{10}}-\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}}{9 e^7 (a+b x) (d+e x)^9}+\frac{b^4 (6 b B d-A b e-5 a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}\\ \end{align*}
Mathematica [A] time = 0.232856, size = 471, normalized size = 1.08 \[ -\frac{\sqrt{(a+b x)^2} \left (28 a^2 b^3 e^2 \left (9 A e \left (13 d^2 e x+d^3+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (78 d^2 e^2 x^2+13 d^3 e x+d^4+286 d e^3 x^3+715 e^4 x^4\right )\right )+84 a^3 b^2 e^3 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (13 d^2 e x+d^3+78 d e^2 x^2+286 e^3 x^3\right )\right )+210 a^4 b e^4 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+462 a^5 e^5 (12 A e+B (d+13 e x))+7 a b^4 e \left (8 A e \left (78 d^2 e^2 x^2+13 d^3 e x+d^4+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (78 d^3 e^2 x^2+286 d^2 e^3 x^3+13 d^4 e x+d^5+715 d e^4 x^4+1287 e^5 x^5\right )\right )+b^5 \left (7 A e \left (78 d^3 e^2 x^2+286 d^2 e^3 x^3+13 d^4 e x+d^5+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+13 d^5 e x+d^6+1287 d e^5 x^5+1716 e^6 x^6\right )\right )\right )}{72072 e^7 (a+b x) (d+e x)^{13}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 689, normalized size = 1.6 \begin{align*} -{\frac{10296\,B{x}^{6}{b}^{5}{e}^{6}+9009\,A{x}^{5}{b}^{5}{e}^{6}+45045\,B{x}^{5}a{b}^{4}{e}^{6}+7722\,B{x}^{5}{b}^{5}d{e}^{5}+40040\,A{x}^{4}a{b}^{4}{e}^{6}+5005\,A{x}^{4}{b}^{5}d{e}^{5}+80080\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+25025\,B{x}^{4}a{b}^{4}d{e}^{5}+4290\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+72072\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+16016\,A{x}^{3}a{b}^{4}d{e}^{5}+2002\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+72072\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+32032\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+10010\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+1716\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+65520\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+19656\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+4368\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+546\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+32760\,B{x}^{2}{a}^{4}b{e}^{6}+19656\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+8736\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+2730\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+468\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+30030\,Ax{a}^{4}b{e}^{6}+10920\,Ax{a}^{3}{b}^{2}d{e}^{5}+3276\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+728\,Axa{b}^{4}{d}^{3}{e}^{3}+91\,Ax{b}^{5}{d}^{4}{e}^{2}+6006\,Bx{a}^{5}{e}^{6}+5460\,Bx{a}^{4}bd{e}^{5}+3276\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+1456\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+455\,Bxa{b}^{4}{d}^{4}{e}^{2}+78\,Bx{b}^{5}{d}^{5}e+5544\,A{a}^{5}{e}^{6}+2310\,Ad{e}^{5}{a}^{4}b+840\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+252\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+56\,Aa{b}^{4}{d}^{4}{e}^{2}+7\,A{b}^{5}{d}^{5}e+462\,Bd{e}^{5}{a}^{5}+420\,B{a}^{4}b{d}^{2}{e}^{4}+252\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+112\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+35\,Ba{b}^{4}{d}^{5}e+6\,B{b}^{5}{d}^{6}}{72072\,{e}^{7} \left ( ex+d \right ) ^{13} \left ( bx+a \right ) ^{5}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.66209, size = 1520, normalized size = 3.49 \begin{align*} -\frac{10296 \, B b^{5} e^{6} x^{6} + 6 \, B b^{5} d^{6} + 5544 \, A a^{5} e^{6} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 420 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 462 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 1287 \,{\left (6 \, B b^{5} d e^{5} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 715 \,{\left (6 \, B b^{5} d^{2} e^{4} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 286 \,{\left (6 \, B b^{5} d^{3} e^{3} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 78 \,{\left (6 \, B b^{5} d^{4} e^{2} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 420 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 13 \,{\left (6 \, B b^{5} d^{5} e + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 420 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 462 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{72072 \,{\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.19059, size = 1241, normalized size = 2.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]