Optimal. Leaf size=193 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{28 e (d+e x)^7 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.142138, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {770, 78, 45, 37} \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{28 e (d+e x)^7 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 78
Rule 45
Rule 37
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)^9} \, 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)^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{(B d-A e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e (b d-a e) (d+e x)^8}+\frac{\left ((3 b B d+A b e-4 a B e) \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^8} \, dx}{4 b^4 e (b d-a e) \left (a b+b^2 x\right )}\\ &=-\frac{(B d-A e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e (b d-a e) (d+e x)^8}+\frac{(3 b B d+A b e-4 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 e (b d-a e)^2 (d+e x)^7}+\frac{\left ((3 b B d+A b e-4 a B e) \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{28 b^3 e (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=-\frac{(B d-A e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e (b d-a e) (d+e x)^8}+\frac{(3 b B d+A b e-4 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 e (b d-a e)^2 (d+e x)^7}+\frac{b (3 b B d+A b e-4 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{168 e (b d-a e)^3 (d+e x)^6}\\ \end{align*}
Mathematica [B] time = 0.235831, size = 466, normalized size = 2.41 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 b^3 e^2 \left (A e \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+B \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )\right )+2 a^3 b^2 e^3 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )\right )+5 a^4 b e^4 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+3 a^5 e^5 (7 A e+B (d+8 e x))+a b^4 e \left (3 A e \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )\right )+b^5 \left (A e \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+8 d^5 e x+d^6+56 d e^5 x^5+28 e^6 x^6\right )\right )\right )}{168 e^7 (a+b x) (d+e x)^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.009, size = 688, normalized size = 3.6 \begin{align*} -{\frac{84\,B{x}^{6}{b}^{5}{e}^{6}+56\,A{x}^{5}{b}^{5}{e}^{6}+280\,B{x}^{5}a{b}^{4}{e}^{6}+168\,B{x}^{5}{b}^{5}d{e}^{5}+210\,A{x}^{4}a{b}^{4}{e}^{6}+70\,A{x}^{4}{b}^{5}d{e}^{5}+420\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+350\,B{x}^{4}a{b}^{4}d{e}^{5}+210\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+336\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+168\,A{x}^{3}a{b}^{4}d{e}^{5}+56\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+336\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+336\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+280\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+168\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+280\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+168\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+84\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+28\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+140\,B{x}^{2}{a}^{4}b{e}^{6}+168\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+168\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+140\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+84\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+120\,Ax{a}^{4}b{e}^{6}+80\,Ax{a}^{3}{b}^{2}d{e}^{5}+48\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+24\,Axa{b}^{4}{d}^{3}{e}^{3}+8\,Ax{b}^{5}{d}^{4}{e}^{2}+24\,Bx{a}^{5}{e}^{6}+40\,Bx{a}^{4}bd{e}^{5}+48\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+48\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+40\,Bxa{b}^{4}{d}^{4}{e}^{2}+24\,Bx{b}^{5}{d}^{5}e+21\,A{a}^{5}{e}^{6}+15\,Ad{e}^{5}{a}^{4}b+10\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+6\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+3\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+3\,Bd{e}^{5}{a}^{5}+5\,B{a}^{4}b{d}^{2}{e}^{4}+6\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+6\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+3\,B{b}^{5}{d}^{6}}{168\,{e}^{7} \left ( ex+d \right ) ^{8} \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 = 1.67667, size = 1305, normalized size = 6.76 \begin{align*} -\frac{84 \, B b^{5} e^{6} x^{6} + 3 \, B b^{5} d^{6} + 21 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 6 \,{\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} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 56 \,{\left (3 \, B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 70 \,{\left (3 \, B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 56 \,{\left (3 \, B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 6 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 28 \,{\left (3 \, B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 6 \,{\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} + 8 \,{\left (3 \, B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 3 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 6 \,{\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} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{168 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} 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 = 1.21051, size = 1239, normalized size = 6.42 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]