Optimal. Leaf size=309 \[ -\frac {2 c^{3/2} \sqrt {d} \sqrt {a+b x^2} (2 b c-3 a d) \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a^3 \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {c+d x^2} (4 b c-3 a d)}{15 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac {\sqrt {c+d x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} \sqrt {b} \sqrt {a+b x^2} (b c-a d)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {x \sqrt {c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {412, 527, 525, 418, 411} \[ \frac {\sqrt {c+d x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} \sqrt {b} \sqrt {a+b x^2} (b c-a d)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {2 c^{3/2} \sqrt {d} \sqrt {a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^3 \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {c+d x^2} (4 b c-3 a d)}{15 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac {x \sqrt {c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 412
Rule 418
Rule 525
Rule 527
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{7/2}} \, dx &=\frac {x \sqrt {c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {-4 c-3 d x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx}{5 a}\\ &=\frac {x \sqrt {c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}}+\frac {(4 b c-3 a d) x \sqrt {c+d x^2}}{15 a^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {c (8 b c-9 a d)+d (4 b c-3 a d) x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx}{15 a^2 (b c-a d)}\\ &=\frac {x \sqrt {c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}}+\frac {(4 b c-3 a d) x \sqrt {c+d x^2}}{15 a^2 (b c-a d) \left (a+b x^2\right )^{3/2}}-\frac {(2 c d (2 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a^2 (b c-a d)^2}+\frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^2 (b c-a d)^2}\\ &=\frac {x \sqrt {c+d x^2}}{5 a \left (a+b x^2\right )^{5/2}}+\frac {(4 b c-3 a d) x \sqrt {c+d x^2}}{15 a^2 (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} \sqrt {b} (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {2 c^{3/2} \sqrt {d} (2 b c-3 a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^3 (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.55, size = 285, normalized size = 0.92 \[ \frac {x \sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (\left (a+b x^2\right )^2 \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right )+3 a^2 (b c-a d)^2+a \left (a+b x^2\right ) (a d-b c) (3 a d-4 b c)\right )+i c \sqrt {\frac {b x^2}{a}+1} \left (a+b x^2\right )^2 \sqrt {\frac {d x^2}{c}+1} \left (\left (-9 a^2 d^2+17 a b c d-8 b^2 c^2\right ) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (x \sqrt {\frac {b}{a}}\right ),\frac {a d}{b c}\right )+\left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{15 a^3 \sqrt {\frac {b}{a}} \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1411, normalized size = 4.57 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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