Optimal. Leaf size=196 \[ \frac {x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac {d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac {\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{11/2}}+\frac {x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac {d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac {d^5 x^5}{5 b^3} \]
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Rubi [A] time = 0.23, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {390, 1157, 385, 205} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 390
Rule 1157
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*}
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Mathematica [A] time = 0.13, size = 196, normalized size = 1.00 \begin {gather*} \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^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 (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac {d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac {d^5 x^5}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+d x^2\right )^5}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.92, size = 1044, normalized size = 5.33 \begin {gather*} \left [\frac {48 \, a^{3} b^{5} d^{5} x^{9} + 16 \, {\left (25 \, a^{3} b^{5} c d^{4} - 9 \, a^{4} b^{4} d^{5}\right )} x^{7} + 16 \, {\left (150 \, a^{3} b^{5} c^{2} d^{3} - 175 \, a^{4} b^{4} c d^{4} + 63 \, a^{5} b^{3} d^{5}\right )} x^{5} + 10 \, {\left (9 \, a b^{7} c^{5} + 15 \, a^{2} b^{6} c^{4} d - 150 \, a^{3} b^{5} c^{3} d^{2} + 750 \, a^{4} b^{4} c^{2} d^{3} - 875 \, a^{5} b^{3} c d^{4} + 315 \, a^{6} b^{2} d^{5}\right )} x^{3} + 15 \, {\left (3 \, a^{2} b^{5} c^{5} + 5 \, a^{3} b^{4} c^{4} d + 30 \, a^{4} b^{3} c^{3} d^{2} - 150 \, a^{5} b^{2} c^{2} d^{3} + 175 \, a^{6} b c d^{4} - 63 \, a^{7} d^{5} + {\left (3 \, b^{7} c^{5} + 5 \, a b^{6} c^{4} d + 30 \, a^{2} b^{5} c^{3} d^{2} - 150 \, a^{3} b^{4} c^{2} d^{3} + 175 \, a^{4} b^{3} c d^{4} - 63 \, a^{5} b^{2} d^{5}\right )} x^{4} + 2 \, {\left (3 \, a b^{6} c^{5} + 5 \, a^{2} b^{5} c^{4} d + 30 \, a^{3} b^{4} c^{3} d^{2} - 150 \, a^{4} b^{3} c^{2} d^{3} + 175 \, a^{5} b^{2} c d^{4} - 63 \, a^{6} b d^{5}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (5 \, a^{2} b^{6} c^{5} - 5 \, a^{3} b^{5} c^{4} d - 30 \, a^{4} b^{4} c^{3} d^{2} + 150 \, a^{5} b^{3} c^{2} d^{3} - 175 \, a^{6} b^{2} c d^{4} + 63 \, a^{7} b d^{5}\right )} x}{240 \, {\left (a^{3} b^{8} x^{4} + 2 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}, \frac {24 \, a^{3} b^{5} d^{5} x^{9} + 8 \, {\left (25 \, a^{3} b^{5} c d^{4} - 9 \, a^{4} b^{4} d^{5}\right )} x^{7} + 8 \, {\left (150 \, a^{3} b^{5} c^{2} d^{3} - 175 \, a^{4} b^{4} c d^{4} + 63 \, a^{5} b^{3} d^{5}\right )} x^{5} + 5 \, {\left (9 \, a b^{7} c^{5} + 15 \, a^{2} b^{6} c^{4} d - 150 \, a^{3} b^{5} c^{3} d^{2} + 750 \, a^{4} b^{4} c^{2} d^{3} - 875 \, a^{5} b^{3} c d^{4} + 315 \, a^{6} b^{2} d^{5}\right )} x^{3} + 15 \, {\left (3 \, a^{2} b^{5} c^{5} + 5 \, a^{3} b^{4} c^{4} d + 30 \, a^{4} b^{3} c^{3} d^{2} - 150 \, a^{5} b^{2} c^{2} d^{3} + 175 \, a^{6} b c d^{4} - 63 \, a^{7} d^{5} + {\left (3 \, b^{7} c^{5} + 5 \, a b^{6} c^{4} d + 30 \, a^{2} b^{5} c^{3} d^{2} - 150 \, a^{3} b^{4} c^{2} d^{3} + 175 \, a^{4} b^{3} c d^{4} - 63 \, a^{5} b^{2} d^{5}\right )} x^{4} + 2 \, {\left (3 \, a b^{6} c^{5} + 5 \, a^{2} b^{5} c^{4} d + 30 \, a^{3} b^{4} c^{3} d^{2} - 150 \, a^{4} b^{3} c^{2} d^{3} + 175 \, a^{5} b^{2} c d^{4} - 63 \, a^{6} b d^{5}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (5 \, a^{2} b^{6} c^{5} - 5 \, a^{3} b^{5} c^{4} d - 30 \, a^{4} b^{4} c^{3} d^{2} + 150 \, a^{5} b^{3} c^{2} d^{3} - 175 \, a^{6} b^{2} c d^{4} + 63 \, a^{7} b d^{5}\right )} x}{120 \, {\left (a^{3} b^{8} x^{4} + 2 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 340, normalized size = 1.73 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 484, normalized size = 2.47 \begin {gather*} \frac {17 a^{3} d^{5} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}-\frac {65 a^{2} c \,d^{4} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}+\frac {45 a \,c^{2} d^{3} x^{3}}{4 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {5 c^{4} d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a}+\frac {3 b \,c^{5} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {25 c^{3} d^{2} x^{3}}{4 \left (b \,x^{2}+a \right )^{2} b}+\frac {d^{5} x^{5}}{5 b^{3}}+\frac {15 a^{4} d^{5} x}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {55 a^{3} c \,d^{4} x}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {35 a^{2} c^{2} d^{3} x}{4 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {15 a \,c^{3} d^{2} x}{4 \left (b \,x^{2}+a \right )^{2} b^{2}}-\frac {a \,d^{5} x^{3}}{b^{4}}+\frac {5 c^{5} x}{8 \left (b \,x^{2}+a \right )^{2} a}-\frac {5 c^{4} d x}{8 \left (b \,x^{2}+a \right )^{2} b}+\frac {5 c \,d^{4} x^{3}}{3 b^{3}}-\frac {63 a^{3} d^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{5}}+\frac {175 a^{2} c \,d^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{4}}-\frac {75 a \,c^{2} d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, b^{3}}+\frac {5 c^{4} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b}+\frac {3 c^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}+\frac {15 c^{3} d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, b^{2}}+\frac {6 a^{2} d^{5} x}{b^{5}}-\frac {15 a c \,d^{4} x}{b^{4}}+\frac {10 c^{2} d^{3} x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 334, normalized size = 1.70 \begin {gather*} \frac {{\left (3 \, b^{6} c^{5} + 5 \, a b^{5} c^{4} d - 50 \, a^{2} b^{4} c^{3} d^{2} + 90 \, a^{3} b^{3} c^{2} d^{3} - 65 \, a^{4} b^{2} c d^{4} + 17 \, a^{5} b d^{5}\right )} x^{3} + 5 \, {\left (a b^{5} c^{5} - a^{2} b^{4} c^{4} d - 6 \, a^{3} b^{3} c^{3} d^{2} + 14 \, a^{4} b^{2} c^{2} d^{3} - 11 \, a^{5} b c d^{4} + 3 \, a^{6} d^{5}\right )} x}{8 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} + \frac {3 \, b^{2} d^{5} x^{5} + 5 \, {\left (5 \, b^{2} c d^{4} - 3 \, a b d^{5}\right )} x^{3} + 15 \, {\left (10 \, b^{2} c^{2} d^{3} - 15 \, a b c d^{4} + 6 \, a^{2} d^{5}\right )} x}{15 \, b^{5}} + \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.02, size = 409, normalized size = 2.09 \begin {gather*} \frac {\frac {5\,x\,\left (3\,a^5\,d^5-11\,a^4\,b\,c\,d^4+14\,a^3\,b^2\,c^2\,d^3-6\,a^2\,b^3\,c^3\,d^2-a\,b^4\,c^4\,d+b^5\,c^5\right )}{8\,a}+\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 )}{8\,a^2}}{a^2\,b^5+2\,a\,b^6\,x^2+b^7\,x^4}-x^3\,\left (\frac {a\,d^5}{b^4}-\frac {5\,c\,d^4}{3\,b^3}\right )+x\,\left (\frac {3\,a\,\left (\frac {3\,a\,d^5}{b^4}-\frac {5\,c\,d^4}{b^3}\right )}{b}-\frac {3\,a^2\,d^5}{b^5}+\frac {10\,c^2\,d^3}{b^3}\right )+\frac {d^5\,x^5}{5\,b^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3\,\left (63\,a^2\,d^2+14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{\sqrt {a}\,\left (-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\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (63\,a^2\,d^2+14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,a^{5/2}\,b^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.41, size = 615, normalized size = 3.14 \begin {gather*} x^{3} \left (- \frac {a d^{5}}{b^{4}} + \frac {5 c d^{4}}{3 b^{3}}\right ) + x \left (\frac {6 a^{2} d^{5}}{b^{5}} - \frac {15 a c d^{4}}{b^{4}} + \frac {10 c^{2} d^{3}}{b^{3}}\right ) + \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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