Optimal. Leaf size=392 \[ -\frac{256 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{315 a^9 \sqrt{a+b x^2}}-\frac{128 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{32 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{16 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.546361, antiderivative size = 380, normalized size of antiderivative = 0.97, number of steps used = 10, 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} \[ -\frac{256 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^9 \sqrt{a+b x^2}}-\frac{128 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{32 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{16 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}+\frac{-15 a^3 D-36 a b (2 b B-a C)+128 A b^3}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}} \]
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^{10} \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{16 A b-9 a \left (B+C x^2+D x^4\right )}{x^8 \left (a+b x^2\right )^{9/2}} \, dx}{9 a}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}+\frac{\int \frac{14 b (16 A b-9 a B)-7 a \left (-9 a C-9 a D x^2\right )}{x^6 \left (a+b x^2\right )^{9/2}} \, dx}{63 a^2}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{12 b \left (224 A b^2-126 a b B+63 a^2 C\right )-315 a^3 D}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{315 a^3}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}-\frac{\left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) \int \frac{1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}+\frac{\left (2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{9 a^4}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{\left (16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{9/2}} \, dx}{9 a^5}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{\left (32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{7/2}} \, dx}{21 a^6}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{\left (128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{105 a^7}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{\left (256 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{315 a^8}\\ &=-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac{16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{256 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^9 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.162759, size = 270, normalized size = 0.69 \[ \frac{256 a^3 b^5 x^{10} \left (-280 A+315 B x^2-126 C x^4+15 D x^6\right )+4480 a^4 b^4 x^8 \left (-2 A+9 B x^2-9 C x^4+3 D x^6\right )+112 a^5 b^3 x^6 \left (8 A+45 B x^2-180 C x^4+150 D x^6\right )-56 a^6 b^2 x^4 \left (4 A+9 B x^2+45 C x^4-150 D x^6\right )-1024 a^2 b^6 x^{12} \left (140 A-63 B x^2+9 C x^4\right )+2 a^7 b x^2 \left (40 A+21 \left (3 B x^2+6 C x^4+25 D x^6\right )\right )-a^8 \left (35 A+45 B x^2+63 C x^4+105 D x^6\right )+2048 a b^7 x^{14} \left (9 B x^2-56 A\right )-32768 A b^8 x^{16}}{315 a^9 x^9 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 349, normalized size = 0.9 \begin{align*} -{\frac{32768\,A{b}^{8}{x}^{16}-18432\,Ba{b}^{7}{x}^{16}+9216\,C{a}^{2}{b}^{6}{x}^{16}-3840\,D{a}^{3}{b}^{5}{x}^{16}+114688\,Aa{b}^{7}{x}^{14}-64512\,B{a}^{2}{b}^{6}{x}^{14}+32256\,C{a}^{3}{b}^{5}{x}^{14}-13440\,D{a}^{4}{b}^{4}{x}^{14}+143360\,A{a}^{2}{b}^{6}{x}^{12}-80640\,B{a}^{3}{b}^{5}{x}^{12}+40320\,C{a}^{4}{b}^{4}{x}^{12}-16800\,D{a}^{5}{b}^{3}{x}^{12}+71680\,A{a}^{3}{b}^{5}{x}^{10}-40320\,B{a}^{4}{b}^{4}{x}^{10}+20160\,C{a}^{5}{b}^{3}{x}^{10}-8400\,D{a}^{6}{b}^{2}{x}^{10}+8960\,A{a}^{4}{b}^{4}{x}^{8}-5040\,B{a}^{5}{b}^{3}{x}^{8}+2520\,C{a}^{6}{b}^{2}{x}^{8}-1050\,D{a}^{7}b{x}^{8}-896\,A{a}^{5}{b}^{3}{x}^{6}+504\,B{a}^{6}{b}^{2}{x}^{6}-252\,C{a}^{7}b{x}^{6}+105\,D{a}^{8}{x}^{6}+224\,A{a}^{6}{b}^{2}{x}^{4}-126\,B{a}^{7}b{x}^{4}+63\,C{a}^{8}{x}^{4}-80\,A{a}^{7}b{x}^{2}+45\,B{a}^{8}{x}^{2}+35\,A{a}^{8}}{315\,{x}^{9}{a}^{9}} \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 = 1.32324, size = 1569, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]