Optimal. Leaf size=193 \[ \frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt {e} \sqrt {c+d x^2}}-\frac {2 b \sqrt {e x} \sqrt {c+d x^2} (5 b c-14 a d)}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3} \]
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Rubi [A] time = 0.16, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {464, 459, 329, 220} \[ \frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt {e} \sqrt {c+d x^2}}-\frac {2 b \sqrt {e x} \sqrt {c+d x^2} (5 b c-14 a d)}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 459
Rule 464
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx &=\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {2 \int \frac {\frac {7 a^2 d}{2}-\frac {1}{2} b (5 b c-14 a d) x^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{7 d}\\ &=-\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}-\frac {1}{21} \left (-21 a^2-\frac {b c (5 b c-14 a d)}{d^2}\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx\\ &=-\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (2 \left (21 a^2+\frac {b c (5 b c-14 a d)}{d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 e}\\ &=-\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (21 a^2+\frac {b c (5 b c-14 a d)}{d^2}\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} \sqrt [4]{d} \sqrt {e} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.25, size = 148, normalized size = 0.77 \[ \frac {2 x \left (-b \left (c+d x^2\right ) \left (-14 a d+5 b c-3 b d x^2\right )+\frac {i \sqrt {x} \sqrt {\frac {c}{d x^2}+1} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right )\right |-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 d^2 \sqrt {e x} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d e x^{3} + c e x}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 350, normalized size = 1.81 \[ \frac {6 b^{2} d^{3} x^{5}+28 a b \,d^{3} x^{3}-4 b^{2} c \,d^{2} x^{3}+28 a b c \,d^{2} x -10 b^{2} c^{2} d x +21 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a^{2} d^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-14 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a b c d \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+5 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, b^{2} c^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{21 \sqrt {d \,x^{2}+c}\, \sqrt {e x}\, d^{3}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^2}{\sqrt {e\,x}\,\sqrt {d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.31, size = 144, normalized size = 0.75 \[ \frac {a^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {a b x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \sqrt {e} \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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