Optimal. Leaf size=224 \[ \frac {16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt {c+d x^2} (b c-a d)}+\frac {8 a^2 x \left (a+b x^2\right ) (9 b c-8 a d)}{315 c^4 \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {2 a x \left (a+b x^2\right )^2 (9 b c-8 a d)}{105 c^3 \left (c+d x^2\right )^{5/2} (b c-a d)}+\frac {x \left (a+b x^2\right )^3 (9 b c-8 a d)}{63 c^2 \left (c+d x^2\right )^{7/2} (b c-a d)}-\frac {d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)} \]
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Rubi [A] time = 0.10, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {382, 378, 191} \begin {gather*} \frac {8 a^2 x \left (a+b x^2\right ) (9 b c-8 a d)}{315 c^4 \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt {c+d x^2} (b c-a d)}+\frac {x \left (a+b x^2\right )^3 (9 b c-8 a d)}{63 c^2 \left (c+d x^2\right )^{7/2} (b c-a d)}+\frac {2 a x \left (a+b x^2\right )^2 (9 b c-8 a d)}{105 c^3 \left (c+d x^2\right )^{5/2} (b c-a d)}-\frac {d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 378
Rule 382
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx &=-\frac {d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac {(9 b c-8 a d) \int \frac {\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{9/2}} \, dx}{9 c (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac {(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {(2 a (9 b c-8 a d)) \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx}{21 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac {(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {2 a (9 b c-8 a d) x \left (a+b x^2\right )^2}{105 c^3 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {\left (8 a^2 (9 b c-8 a d)\right ) \int \frac {a+b x^2}{\left (c+d x^2\right )^{5/2}} \, dx}{105 c^3 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac {(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {2 a (9 b c-8 a d) x \left (a+b x^2\right )^2}{105 c^3 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {8 a^2 (9 b c-8 a d) x \left (a+b x^2\right )}{315 c^4 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {\left (16 a^3 (9 b c-8 a d)\right ) \int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx}{315 c^4 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^4}{9 c (b c-a d) \left (c+d x^2\right )^{9/2}}+\frac {(9 b c-8 a d) x \left (a+b x^2\right )^3}{63 c^2 (b c-a d) \left (c+d x^2\right )^{7/2}}+\frac {2 a (9 b c-8 a d) x \left (a+b x^2\right )^2}{105 c^3 (b c-a d) \left (c+d x^2\right )^{5/2}}+\frac {8 a^2 (9 b c-8 a d) x \left (a+b x^2\right )}{315 c^4 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {16 a^3 (9 b c-8 a d) x}{315 c^5 (b c-a d) \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 163, normalized size = 0.73 \begin {gather*} \frac {a^3 \left (315 c^4 x+840 c^3 d x^3+1008 c^2 d^2 x^5+576 c d^3 x^7+128 d^4 x^9\right )+3 a^2 b c x^3 \left (105 c^3+126 c^2 d x^2+72 c d^2 x^4+16 d^3 x^6\right )+3 a b^2 c^2 x^5 \left (63 c^2+36 c d x^2+8 d^2 x^4\right )+5 b^3 c^3 x^7 \left (9 c+2 d x^2\right )}{315 c^5 \left (c+d x^2\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 193, normalized size = 0.86 \begin {gather*} \frac {315 a^3 c^4 x+840 a^3 c^3 d x^3+1008 a^3 c^2 d^2 x^5+576 a^3 c d^3 x^7+128 a^3 d^4 x^9+315 a^2 b c^4 x^3+378 a^2 b c^3 d x^5+216 a^2 b c^2 d^2 x^7+48 a^2 b c d^3 x^9+189 a b^2 c^4 x^5+108 a b^2 c^3 d x^7+24 a b^2 c^2 d^2 x^9+45 b^3 c^4 x^7+10 b^3 c^3 d x^9}{315 c^5 \left (c+d x^2\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.62, size = 229, normalized size = 1.02 \begin {gather*} \frac {{\left (2 \, {\left (5 \, b^{3} c^{3} d + 12 \, a b^{2} c^{2} d^{2} + 24 \, a^{2} b c d^{3} + 64 \, a^{3} d^{4}\right )} x^{9} + 315 \, a^{3} c^{4} x + 9 \, {\left (5 \, b^{3} c^{4} + 12 \, a b^{2} c^{3} d + 24 \, a^{2} b c^{2} d^{2} + 64 \, a^{3} c d^{3}\right )} x^{7} + 63 \, {\left (3 \, a b^{2} c^{4} + 6 \, a^{2} b c^{3} d + 16 \, a^{3} c^{2} d^{2}\right )} x^{5} + 105 \, {\left (3 \, a^{2} b c^{4} + 8 \, a^{3} c^{3} d\right )} x^{3}\right )} \sqrt {d x^{2} + c}}{315 \, {\left (c^{5} d^{5} x^{10} + 5 \, c^{6} d^{4} x^{8} + 10 \, c^{7} d^{3} x^{6} + 10 \, c^{8} d^{2} x^{4} + 5 \, c^{9} d x^{2} + c^{10}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 218, normalized size = 0.97 \begin {gather*} \frac {{\left ({\left ({\left (x^{2} {\left (\frac {2 \, {\left (5 \, b^{3} c^{3} d^{5} + 12 \, a b^{2} c^{2} d^{6} + 24 \, a^{2} b c d^{7} + 64 \, a^{3} d^{8}\right )} x^{2}}{c^{5} d^{4}} + \frac {9 \, {\left (5 \, b^{3} c^{4} d^{4} + 12 \, a b^{2} c^{3} d^{5} + 24 \, a^{2} b c^{2} d^{6} + 64 \, a^{3} c d^{7}\right )}}{c^{5} d^{4}}\right )} + \frac {63 \, {\left (3 \, a b^{2} c^{4} d^{4} + 6 \, a^{2} b c^{3} d^{5} + 16 \, a^{3} c^{2} d^{6}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac {105 \, {\left (3 \, a^{2} b c^{4} d^{4} + 8 \, a^{3} c^{3} d^{5}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac {315 \, a^{3}}{c}\right )} x}{315 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 190, normalized size = 0.85 \begin {gather*} \frac {\left (128 a^{3} d^{4} x^{8}+48 a^{2} b c \,d^{3} x^{8}+24 a \,b^{2} c^{2} d^{2} x^{8}+10 b^{3} c^{3} d \,x^{8}+576 a^{3} c \,d^{3} x^{6}+216 a^{2} b \,c^{2} d^{2} x^{6}+108 a \,b^{2} c^{3} d \,x^{6}+45 b^{3} c^{4} x^{6}+1008 a^{3} c^{2} d^{2} x^{4}+378 a^{2} b \,c^{3} d \,x^{4}+189 a \,b^{2} c^{4} x^{4}+840 a^{3} c^{3} d \,x^{2}+315 a^{2} b \,c^{4} x^{2}+315 a^{3} c^{4}\right ) x}{315 \left (d \,x^{2}+c \right )^{\frac {9}{2}} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.51, size = 465, normalized size = 2.08 \begin {gather*} -\frac {b^{3} x^{5}}{4 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d} - \frac {5 \, b^{3} c x^{3}}{24 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d^{2}} - \frac {a b^{2} x^{3}}{2 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d} + \frac {128 \, a^{3} x}{315 \, \sqrt {d x^{2} + c} c^{5}} + \frac {64 \, a^{3} x}{315 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{4}} + \frac {16 \, a^{3} x}{105 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c^{3}} + \frac {8 \, a^{3} x}{63 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} c^{2}} + \frac {a^{3} x}{9 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} c} + \frac {b^{3} x}{84 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} d^{3}} + \frac {2 \, b^{3} x}{63 \, \sqrt {d x^{2} + c} c^{2} d^{3}} + \frac {b^{3} x}{63 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c d^{3}} + \frac {5 \, b^{3} c x}{504 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} d^{3}} - \frac {5 \, b^{3} c^{2} x}{72 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d^{3}} + \frac {a b^{2} x}{42 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} d^{2}} + \frac {8 \, a b^{2} x}{105 \, \sqrt {d x^{2} + c} c^{3} d^{2}} + \frac {4 \, a b^{2} x}{105 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2} d^{2}} + \frac {a b^{2} x}{35 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c d^{2}} - \frac {a b^{2} c x}{6 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d^{2}} - \frac {a^{2} b x}{3 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} d} + \frac {16 \, a^{2} b x}{105 \, \sqrt {d x^{2} + c} c^{4} d} + \frac {8 \, a^{2} b x}{105 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{3} d} + \frac {2 \, a^{2} b x}{35 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} c^{2} d} + \frac {a^{2} b x}{21 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.15, size = 326, normalized size = 1.46 \begin {gather*} \frac {x\,\left (\frac {a^3}{9\,c}-\frac {c\,\left (\frac {c\,\left (\frac {b^3}{9\,d}-\frac {a\,b^2}{3\,c}\right )}{d}+\frac {a^2\,b}{3\,c}\right )}{d}\right )}{{\left (d\,x^2+c\right )}^{9/2}}-\frac {x\,\left (\frac {b^3}{5\,d^3}-\frac {16\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-4\,b^3\,c^3}{105\,c^3\,d^3}\right )}{{\left (d\,x^2+c\right )}^{5/2}}+\frac {x\,\left (\frac {c\,\left (\frac {b^3}{7\,d^2}-\frac {b^2\,\left (3\,a\,d-b\,c\right )}{7\,c\,d^2}\right )}{d}+\frac {8\,a^3\,d^3+3\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d+b^3\,c^3}{63\,c^2\,d^3}\right )}{{\left (d\,x^2+c\right )}^{7/2}}+\frac {x\,\left (64\,a^3\,d^3+24\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{315\,c^4\,d^3\,{\left (d\,x^2+c\right )}^{3/2}}+\frac {x\,\left (128\,a^3\,d^3+48\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+10\,b^3\,c^3\right )}{315\,c^5\,d^3\,\sqrt {d\,x^2+c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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