Optimal. Leaf size=445 \[ \frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{15 c d^3 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b \sqrt {c} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {424, 542, 545,
429, 506, 422} \begin {gather*} \frac {b \sqrt {c} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}-\frac {\sqrt {a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 \sqrt {c} d^{7/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt {c+d x^2}}+\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 424
Rule 429
Rule 506
Rule 542
Rule 545
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx &=-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}+\frac {\int \frac {\left (a+b x^2\right )^{3/2} \left (a b c+b (6 b c-5 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{c d}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {\int \frac {\sqrt {a+b x^2} \left (-2 a b c (3 b c-5 a d)-b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x^2\right )}{\sqrt {c+d x^2}} \, dx}{5 c d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {\int \frac {a b c \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right )+b \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 c d^3}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {\left (a b \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 d^3}+\frac {\left (b \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 c d^3}\\ &=\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{15 c d^3 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}+\frac {b \sqrt {c} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \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 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 d^3}\\ &=\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{15 c d^3 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b \sqrt {c} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \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 d^{7/2} \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] Result contains complex when optimal does not.
time = 5.49, size = 318, normalized size = 0.71 \begin {gather*} \frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (-45 a^2 b c d^2+15 a^3 d^3+a b^2 c d \left (61 c+16 d x^2\right )-3 b^3 c \left (8 c^2+2 c d x^2-d^2 x^4\right )\right )+i b c \left (-48 b^3 c^3+128 a b^2 c^2 d-103 a^2 b c d^2+15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+4 i b c \left (12 b^3 c^3-38 a b^2 c^2 d+41 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} c d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 755, normalized size = 1.70 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {7}{2}}}{\left (c + d x^{2}\right )^{\frac {3}{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 )}^{7/2}}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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