Optimal. Leaf size=196 \[ \frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^5 x^5}{5 b^3} \]
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Rubi [A] time = 0.226919, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {390, 1157, 385, 205} \[ \frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^5 x^5}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 390
Rule 1157
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^5}{\left (a+b x^2\right )^3} \, dx &=\int \left (\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right )}{b^5}+\frac{d^4 (5 b c-3 a d) x^2}{b^4}+\frac{d^5 x^4}{b^3}+\frac{(b c-a d)^3 \left (b^2 c^2+3 a b c d+6 a^2 d^2\right )+5 b d (b c-a d)^3 (b c+3 a d) x^2+10 b^2 d^2 (b c-a d)^3 x^4}{b^5 \left (a+b x^2\right )^3}\right ) \, dx\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{\int \frac{(b c-a d)^3 \left (b^2 c^2+3 a b c d+6 a^2 d^2\right )+5 b d (b c-a d)^3 (b c+3 a d) x^2+10 b^2 d^2 (b c-a d)^3 x^4}{\left (a+b x^2\right )^3} \, dx}{b^5}\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}-\frac{\int \frac{-(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+23 a^2 d^2\right )-40 a b d^2 (b c-a d)^3 x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b^5}\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac{(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^2 b^5}\\ &=\frac{d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac{d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac{d^5 x^5}{5 b^3}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac{(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.122091, size = 196, normalized size = 1. \[ \frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^5 x^5}{5 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 484, normalized size = 2.5 \begin{align*}{\frac{{d}^{5}{x}^{5}}{5\,{b}^{3}}}-{\frac{{d}^{5}{x}^{3}a}{{b}^{4}}}+{\frac{5\,{d}^{4}{x}^{3}c}{3\,{b}^{3}}}+6\,{\frac{{a}^{2}{d}^{5}x}{{b}^{5}}}-15\,{\frac{ac{d}^{4}x}{{b}^{4}}}+10\,{\frac{{c}^{2}{d}^{3}x}{{b}^{3}}}+{\frac{17\,{a}^{3}{x}^{3}{d}^{5}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{65\,{a}^{2}{x}^{3}c{d}^{4}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{45\,a{x}^{3}{c}^{2}{d}^{3}}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{25\,{x}^{3}{c}^{3}{d}^{2}}{4\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{x}^{3}{c}^{4}d}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,b{x}^{3}{c}^{5}}{8\, \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}+{\frac{15\,x{a}^{4}{d}^{5}}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{55\,{a}^{3}cx{d}^{4}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{a}^{2}{c}^{2}x{d}^{3}}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,a{c}^{3}x{d}^{2}}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,x{c}^{4}d}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,x{c}^{5}}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{63\,{a}^{3}{d}^{5}}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{175\,{a}^{2}c{d}^{4}}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{75\,a{c}^{2}{d}^{3}}{4\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{15\,{c}^{3}{d}^{2}}{4\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{c}^{4}d}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{5}}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 [B] time = 1.60851, size = 2195, normalized size = 11.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.5243, size = 614, normalized size = 3.13 \begin{align*} \frac{\sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- \frac{a^{3} b^{5} \sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log{\left (\frac{a^{3} b^{5} \sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} + \frac{x^{3} \left (17 a^{5} b d^{5} - 65 a^{4} b^{2} c d^{4} + 90 a^{3} b^{3} c^{2} d^{3} - 50 a^{2} b^{4} c^{3} d^{2} + 5 a b^{5} c^{4} d + 3 b^{6} c^{5}\right ) + x \left (15 a^{6} d^{5} - 55 a^{5} b c d^{4} + 70 a^{4} b^{2} c^{2} d^{3} - 30 a^{3} b^{3} c^{3} d^{2} - 5 a^{2} b^{4} c^{4} d + 5 a b^{5} c^{5}\right )}{8 a^{4} b^{5} + 16 a^{3} b^{6} x^{2} + 8 a^{2} b^{7} x^{4}} + \frac{d^{5} x^{5}}{5 b^{3}} - \frac{x^{3} \left (3 a d^{5} - 5 b c d^{4}\right )}{3 b^{4}} + \frac{x \left (6 a^{2} d^{5} - 15 a b c d^{4} + 10 b^{2} c^{2} d^{3}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13319, size = 459, normalized size = 2.34 \begin{align*} \frac{{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{5}} + \frac{3 \, b^{6} c^{5} x^{3} + 5 \, a b^{5} c^{4} d x^{3} - 50 \, a^{2} b^{4} c^{3} d^{2} x^{3} + 90 \, a^{3} b^{3} c^{2} d^{3} x^{3} - 65 \, a^{4} b^{2} c d^{4} x^{3} + 17 \, a^{5} b d^{5} x^{3} + 5 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{4} d x - 30 \, a^{3} b^{3} c^{3} d^{2} x + 70 \, a^{4} b^{2} c^{2} d^{3} x - 55 \, a^{5} b c d^{4} x + 15 \, a^{6} d^{5} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{5}} + \frac{3 \, b^{12} d^{5} x^{5} + 25 \, b^{12} c d^{4} x^{3} - 15 \, a b^{11} d^{5} x^{3} + 150 \, b^{12} c^{2} d^{3} x - 225 \, a b^{11} c d^{4} x + 90 \, a^{2} b^{10} d^{5} x}{15 \, b^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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