Optimal. Leaf size=340 \[ \frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} \sqrt {a d-b c} (2 a d+5 b c) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{5/2} x}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} \sqrt {a d-b c} (2 a d+5 b c) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{5/2} x}+\frac {b x (5 b c-a d)}{2 c d^2 \sqrt [4]{a+b x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} (5 b c-a d) E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d^2 \sqrt [4]{a+b x^2}}-\frac {x \left (a+b x^2\right )^{3/4} (b c-a d)}{2 c d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.29, antiderivative size = 340, 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 = {413, 530, 229, 227, 196, 399, 490, 1218} \[ \frac {b x (5 b c-a d)}{2 c d^2 \sqrt [4]{a+b x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} (5 b c-a d) E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d^2 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} \sqrt {a d-b c} (2 a d+5 b c) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{5/2} x}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} \sqrt {a d-b c} (2 a d+5 b c) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c d^{5/2} x}-\frac {x \left (a+b x^2\right )^{3/4} (b c-a d)}{2 c d \left (c+d x^2\right )} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rule 229
Rule 399
Rule 413
Rule 490
Rule 530
Rule 1218
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{7/4}}{\left (c+d x^2\right )^2} \, dx &=-\frac {(b c-a d) x \left (a+b x^2\right )^{3/4}}{2 c d \left (c+d x^2\right )}+\frac {\int \frac {a (b c+a d)+\frac {1}{2} b (5 b c-a d) x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )^{3/4}}{2 c d \left (c+d x^2\right )}+\frac {(b (5 b c-a d)) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{4 c d^2}-\frac {((b c-a d) (5 b c+2 a d)) \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )^{3/4}}{2 c d \left (c+d x^2\right )}-\frac {\left ((b c-a d) (5 b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \operatorname {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^2 x}+\frac {\left (b (5 b c-a d) \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{4 c d^2 \sqrt [4]{a+b x^2}}\\ &=\frac {b (5 b c-a d) x}{2 c d^2 \sqrt [4]{a+b x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/4}}{2 c d \left (c+d x^2\right )}+\frac {\left ((b c-a d) (5 b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \operatorname {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^{5/2} x}-\frac {\left ((b c-a d) (5 b c+2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \operatorname {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^{5/2} x}-\frac {\left (b (5 b c-a d) \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^2 \sqrt [4]{a+b x^2}}\\ &=\frac {b (5 b c-a d) x}{2 c d^2 \sqrt [4]{a+b x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/4}}{2 c d \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} (5 b c-a d) \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^2 \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {-b c+a d} (5 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^{5/2} x}-\frac {\sqrt [4]{a} \sqrt {-b c+a d} (5 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^{5/2} x}\\ \end {align*}
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Mathematica [C] time = 0.34, size = 340, normalized size = 1.00 \[ \frac {x \left (\frac {6 c \left (x^2 \left (a+b x^2\right ) (a d-b c) \left (4 a d F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )-6 a c \left (2 a^2 d+a b d x^2-b^2 c x^2\right ) F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}{\left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )-6 a c F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}-b x^2 \sqrt [4]{\frac {b x^2}{a}+1} (a d-5 b c) F_1\left (\frac {3}{2};\frac {1}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}{12 c^2 d \sqrt [4]{a+b x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {7}{4}}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {7}{4}}}{\left (d \,x^{2}+c \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {7}{4}}}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{7/4}}{{\left (d\,x^2+c\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right )^{\frac {7}{4}}}{\left (c + d x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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