Optimal. Leaf size=309 \[ -\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x}-\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x} \]
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Rubi [A]
time = 0.15, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {423, 544, 235,
233, 202, 408, 504, 1232} \begin {gather*} \frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+b c) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+b c) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} x \sqrt {a d-b c}}+\frac {\sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}}-\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 235
Rule 408
Rule 423
Rule 504
Rule 544
Rule 1232
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx &=\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac {\int \frac {-a+\frac {b x^2}{2}}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c}\\ &=\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac {b \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{4 c d}+\frac {(b c+2 a d) \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c d}\\ &=\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac {\left ((b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{a}} \left (b c-a d+d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c d x}-\frac {\left (b \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{4 c d \sqrt [4]{a+b x^2}}\\ &=-\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac {\left ((b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}-\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c d^{3/2} x}+\frac {\left ((b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}+\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c d^{3/2} x}+\frac {\left (b \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{4 c d \sqrt [4]{a+b x^2}}\\ &=-\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x}-\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.15, size = 232, normalized size = 0.75 \begin {gather*} \frac {x \left (-\frac {b x^2 \sqrt [4]{1+\frac {b x^2}{a}} F_1\left (\frac {3}{2};\frac {1}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^2}+\frac {6 \left (\frac {a+b x^2}{c}-\frac {6 a^2 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{-6 a c F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}\right )}{c+d x^2}\right )}{12 \sqrt [4]{a+b x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}}}{\left (d \,x^{2}+c \right )^{2}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{\frac {3}{4}}}{\left (c + d x^{2}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{3/4}}{{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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