Optimal. Leaf size=113 \[ -\frac {8 d^2 \sqrt {c+d x^2}}{15 \sqrt {a+b x^2} (b c-a d)^3}+\frac {4 d \sqrt {c+d x^2}}{15 \left (a+b x^2\right )^{3/2} (b c-a d)^2}-\frac {\sqrt {c+d x^2}}{5 \left (a+b x^2\right )^{5/2} (b c-a d)} \]
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Rubi [A] time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {444, 45, 37} \[ -\frac {8 d^2 \sqrt {c+d x^2}}{15 \sqrt {a+b x^2} (b c-a d)^3}+\frac {4 d \sqrt {c+d x^2}}{15 \left (a+b x^2\right )^{3/2} (b c-a d)^2}-\frac {\sqrt {c+d x^2}}{5 \left (a+b x^2\right )^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+d x^2}}{5 (b c-a d) \left (a+b x^2\right )^{5/2}}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{5 (b c-a d)}\\ &=-\frac {\sqrt {c+d x^2}}{5 (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {4 d \sqrt {c+d x^2}}{15 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {\left (4 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{15 (b c-a d)^2}\\ &=-\frac {\sqrt {c+d x^2}}{5 (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {4 d \sqrt {c+d x^2}}{15 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}-\frac {8 d^2 \sqrt {c+d x^2}}{15 (b c-a d)^3 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 83, normalized size = 0.73 \[ -\frac {\sqrt {c+d x^2} \left (15 a^2 d^2-10 a b d \left (c-2 d x^2\right )+b^2 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )\right )}{15 \left (a+b x^2\right )^{5/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 259, normalized size = 2.29 \[ -\frac {{\left (8 \, b^{2} d^{2} x^{4} + 3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2} - 4 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{4} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 243, normalized size = 2.15 \[ -\frac {16 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 5 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c + 5 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d + 10 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )} \sqrt {b d} b^{3} d^{2}}{15 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{5} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 113, normalized size = 1.00 \[ \frac {\sqrt {d \,x^{2}+c}\, \left (8 b^{2} d^{2} x^{4}+20 a b \,d^{2} x^{2}-4 b^{2} c d \,x^{2}+15 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right )}{15 \left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (a^{3} d^{3}-3 a^{2} c \,d^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 216, normalized size = 1.91 \[ \frac {\sqrt {b\,x^2+a}\,\left (\frac {15\,a^2\,c\,d^2-10\,a\,b\,c^2\,d+3\,b^2\,c^3}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d^3\,x^6}{15\,b\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,\left (15\,a^2\,d^3+10\,a\,b\,c\,d^2-b^2\,c^2\,d\right )}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,d^2\,x^4\,\left (5\,a\,d+b\,c\right )}{15\,b^2\,{\left (a\,d-b\,c\right )}^3}\right )}{x^6\,\sqrt {d\,x^2+c}+\frac {a^3\,\sqrt {d\,x^2+c}}{b^3}+\frac {3\,a\,x^4\,\sqrt {d\,x^2+c}}{b}+\frac {3\,a^2\,x^2\,\sqrt {d\,x^2+c}}{b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b x^{2}\right )^{\frac {7}{2}} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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