Optimal. Leaf size=174 \[ \frac {(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac {d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3}-\frac {d x \sqrt {c+d x^2} (b c-2 a d)}{2 a b^2}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {413, 528, 523, 217, 206, 377, 205} \begin {gather*} \frac {(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac {d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3}-\frac {d x \sqrt {c+d x^2} (b c-2 a d)}{2 a b^2}+\frac {x \left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 217
Rule 377
Rule 413
Rule 523
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\int \frac {\sqrt {c+d x^2} \left (c (b c+a d)-2 d (b c-2 a d) x^2\right )}{a+b x^2} \, dx}{2 a b}\\ &=-\frac {d (b c-2 a d) x \sqrt {c+d x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\int \frac {2 c \left (b^2 c^2+2 a b c d-2 a^2 d^2\right )+2 a d^2 (5 b c-4 a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{4 a b^2}\\ &=-\frac {d (b c-2 a d) x \sqrt {c+d x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (d^2 (5 b c-4 a d)\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 b^3}+\frac {\left ((b c-a d)^2 (b c+4 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a b^3}\\ &=-\frac {d (b c-2 a d) x \sqrt {c+d x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {\left (d^2 (5 b c-4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 b^3}+\frac {\left ((b c-a d)^2 (b c+4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a b^3}\\ &=-\frac {d (b c-2 a d) x \sqrt {c+d x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac {(b c-a d)^{3/2} (b c+4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac {d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 143, normalized size = 0.82 \begin {gather*} \frac {\frac {(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2}}+d^{3/2} (5 b c-4 a d) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )+b x \sqrt {c+d x^2} \left (\frac {(b c-a d)^2}{a \left (a+b x^2\right )}+d^2\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.80, size = 209, normalized size = 1.20 \begin {gather*} \frac {\sqrt {c+d x^2} \left (2 a^2 d^2 x-2 a b c d x+a b d^2 x^3+b^2 c^2 x\right )}{2 a b^2 \left (a+b x^2\right )}-\frac {\sqrt {b c-a d} \left (-4 a^2 d^2+3 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {a \sqrt {d}-b x \sqrt {c+d x^2}+b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2} b^3}+\frac {\left (4 a d^{5/2}-5 b c d^{3/2}\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.37, size = 1228, normalized size = 7.06
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 407, normalized size = 2.34 \begin {gather*} \frac {\sqrt {d x^{2} + c} d^{2} x}{2 \, b^{2}} - \frac {{\left (5 \, b c d^{\frac {3}{2}} - 4 \, a d^{\frac {5}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, b^{3}} - \frac {{\left (b^{3} c^{3} \sqrt {d} + 2 \, a b^{2} c^{2} d^{\frac {3}{2}} - 7 \, a^{2} b c d^{\frac {5}{2}} + 4 \, a^{3} d^{\frac {7}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a b^{3}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{3} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} d^{\frac {3}{2}} + 5 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {7}{2}} - b^{3} c^{4} \sqrt {d} + 2 \, a b^{2} c^{3} d^{\frac {3}{2}} - a^{2} b c^{2} d^{\frac {5}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 7451, normalized size = 42.82 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^{5/2}}{{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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