Optimal. Leaf size=334 \[ \frac{128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt{a+b x^2}}+\frac{64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.480311, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1803, 12, 271, 192, 191} \[ \frac{128 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^8 \sqrt{a+b x^2}}+\frac{64 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{16 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{8 b x \left (48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )\right )}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (3 a^2 D-10 a b C+24 b^2 B\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1803
Rule 12
Rule 271
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{A+B x^2+C x^4+D x^6}{x^8 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{14 A b-7 a \left (B+C x^2+D x^4\right )}{x^6 \left (a+b x^2\right )^{9/2}} \, dx}{7 a}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}+\frac{\int \frac{12 b (14 A b-7 a B)-5 a \left (-7 a C-7 a D x^2\right )}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{35 a^2}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{10 b \left (168 A b^2-84 a b B+35 a^2 C\right )-105 a^3 D}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{105 a^3}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}-\frac{\left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) \int \frac{1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{\left (8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^4}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{\left (16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^5}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{\left (64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^6}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{\left (128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^7}\\ &=-\frac{A}{7 a x^7 \left (a+b x^2\right )^{7/2}}+\frac{2 A b-a B}{5 a^2 x^5 \left (a+b x^2\right )^{7/2}}-\frac{24 A b^2-a (12 b B-5 a C)}{15 a^3 x^3 \left (a+b x^2\right )^{7/2}}+\frac{48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )}{3 a^4 x \left (a+b x^2\right )^{7/2}}+\frac{8 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{21 a^5 \left (a+b x^2\right )^{7/2}}+\frac{16 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{35 a^6 \left (a+b x^2\right )^{5/2}}+\frac{64 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^7 \left (a+b x^2\right )^{3/2}}+\frac{128 b \left (48 A b^3-a \left (24 b^2 B-10 a b C+3 a^2 D\right )\right ) x}{105 a^8 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.146353, size = 234, normalized size = 0.7 \[ \frac{128 a^3 b^4 x^8 \left (105 A-105 B x^2+35 C x^4-3 D x^6\right )+112 a^4 b^3 x^6 \left (15 A-60 B x^2+50 C x^4-12 D x^6\right )-56 a^5 b^2 x^4 \left (3 A+15 B x^2-50 C x^4+30 D x^6\right )+256 a^2 b^5 x^{10} \left (105 A-42 B x^2+5 C x^4\right )+14 a^6 b x^2 \left (3 A+6 B x^2+25 C x^4-60 D x^6\right )-a^7 \left (15 A+21 B x^2+35 x^4 \left (C+3 D x^2\right )\right )-3072 a b^6 x^{12} \left (B x^2-7 A\right )+6144 A b^7 x^{14}}{105 a^8 x^7 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 301, normalized size = 0.9 \begin{align*} -{\frac{-6144\,A{b}^{7}{x}^{14}+3072\,Ba{b}^{6}{x}^{14}-1280\,C{a}^{2}{b}^{5}{x}^{14}+384\,D{a}^{3}{b}^{4}{x}^{14}-21504\,Aa{b}^{6}{x}^{12}+10752\,B{a}^{2}{b}^{5}{x}^{12}-4480\,C{a}^{3}{b}^{4}{x}^{12}+1344\,D{a}^{4}{b}^{3}{x}^{12}-26880\,A{a}^{2}{b}^{5}{x}^{10}+13440\,B{a}^{3}{b}^{4}{x}^{10}-5600\,C{a}^{4}{b}^{3}{x}^{10}+1680\,D{a}^{5}{b}^{2}{x}^{10}-13440\,A{a}^{3}{b}^{4}{x}^{8}+6720\,B{a}^{4}{b}^{3}{x}^{8}-2800\,C{a}^{5}{b}^{2}{x}^{8}+840\,D{a}^{6}b{x}^{8}-1680\,A{a}^{4}{b}^{3}{x}^{6}+840\,B{a}^{5}{b}^{2}{x}^{6}-350\,C{a}^{6}b{x}^{6}+105\,D{a}^{7}{x}^{6}+168\,A{a}^{5}{b}^{2}{x}^{4}-84\,B{a}^{6}b{x}^{4}+35\,C{a}^{7}{x}^{4}-42\,A{a}^{6}b{x}^{2}+21\,B{a}^{7}{x}^{2}+15\,A{a}^{7}}{105\,{x}^{7}{a}^{8}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 = 1.33665, size = 1266, normalized size = 3.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]