Optimal. Leaf size=438 \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x) (d+e x)^5}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{6 e^7 (a+b x) (d+e x)^6}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{10 e^7 (a+b x) (d+e x)^{10}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4} \]
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Rubi [A] time = 0.38993, antiderivative size = 438, 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} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x) (d+e x)^5}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{6 e^7 (a+b x) (d+e x)^6}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{10 e^7 (a+b x) (d+e x)^{10}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
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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)^{11}} \, 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)^{11}} \, 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)^{11}}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{10}}-\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)^9}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^8}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^7}+\frac{b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^6}+\frac{b^{10} B}{e^6 (d+e x)^5}\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}}{10 e^7 (a+b x) (d+e x)^{10}}+\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}}{9 e^7 (a+b x) (d+e x)^9}-\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}}{8 e^7 (a+b x) (d+e x)^8}+\frac{10 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}}{7 e^7 (a+b x) (d+e x)^7}-\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}}{6 e^7 (a+b x) (d+e x)^6}+\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}}{5 e^7 (a+b x) (d+e x)^5}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.243665, size = 468, normalized size = 1.07 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^2 b^3 e^2 \left (3 A e \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )\right )+10 a^3 b^2 e^3 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )\right )+35 a^4 b e^4 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+28 a^5 e^5 (9 A e+B (d+10 e x))+10 a b^4 e \left (A e \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )+B \left (45 d^3 e^2 x^2+120 d^2 e^3 x^3+10 d^4 e x+d^5+210 d e^4 x^4+252 e^5 x^5\right )\right )+b^5 \left (2 A e \left (45 d^3 e^2 x^2+120 d^2 e^3 x^3+10 d^4 e x+d^5+210 d e^4 x^4+252 e^5 x^5\right )+3 B \left (45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+10 d^5 e x+d^6+252 d e^5 x^5+210 e^6 x^6\right )\right )\right )}{2520 e^7 (a+b x) (d+e x)^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 689, normalized size = 1.6 \begin{align*} -{\frac{630\,B{x}^{6}{b}^{5}{e}^{6}+504\,A{x}^{5}{b}^{5}{e}^{6}+2520\,B{x}^{5}a{b}^{4}{e}^{6}+756\,B{x}^{5}{b}^{5}d{e}^{5}+2100\,A{x}^{4}a{b}^{4}{e}^{6}+420\,A{x}^{4}{b}^{5}d{e}^{5}+4200\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+2100\,B{x}^{4}a{b}^{4}d{e}^{5}+630\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+3600\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+1200\,A{x}^{3}a{b}^{4}d{e}^{5}+240\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+3600\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+2400\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+1200\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+360\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+3150\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+1350\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+450\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+90\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+1575\,B{x}^{2}{a}^{4}b{e}^{6}+1350\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+900\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+450\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+135\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+1400\,Ax{a}^{4}b{e}^{6}+700\,Ax{a}^{3}{b}^{2}d{e}^{5}+300\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+100\,Axa{b}^{4}{d}^{3}{e}^{3}+20\,Ax{b}^{5}{d}^{4}{e}^{2}+280\,Bx{a}^{5}{e}^{6}+350\,Bx{a}^{4}bd{e}^{5}+300\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+200\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+100\,Bxa{b}^{4}{d}^{4}{e}^{2}+30\,Bx{b}^{5}{d}^{5}e+252\,A{a}^{5}{e}^{6}+140\,Ad{e}^{5}{a}^{4}b+70\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+30\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+10\,Aa{b}^{4}{d}^{4}{e}^{2}+2\,A{b}^{5}{d}^{5}e+28\,Bd{e}^{5}{a}^{5}+35\,B{a}^{4}b{d}^{2}{e}^{4}+30\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+20\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+10\,Ba{b}^{4}{d}^{5}e+3\,B{b}^{5}{d}^{6}}{2520\,{e}^{7} \left ( ex+d \right ) ^{10} \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.
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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.
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Fricas [A] time = 1.64928, size = 1411, normalized size = 3.22 \begin{align*} -\frac{630 \, B b^{5} e^{6} x^{6} + 3 \, B b^{5} d^{6} + 252 \, A a^{5} e^{6} + 2 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 10 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 30 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 35 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 28 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 252 \,{\left (3 \, B b^{5} d e^{5} + 2 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 210 \,{\left (3 \, B b^{5} d^{2} e^{4} + 2 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 10 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 120 \,{\left (3 \, B b^{5} d^{3} e^{3} + 2 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 10 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 30 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 45 \,{\left (3 \, B b^{5} d^{4} e^{2} + 2 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 10 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 30 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 35 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 10 \,{\left (3 \, B b^{5} d^{5} e + 2 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 10 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 30 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 35 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 28 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{2520 \,{\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.16892, size = 1241, normalized size = 2.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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