Optimal. Leaf size=346 \[ -\frac {2 b \sqrt {c} \sqrt {a+b x^2} (2 b c-3 a d) \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \sqrt {a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right )}{3 c d^2 \sqrt {c+d x^2}}+\frac {\sqrt {a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-3 a d)}{3 c d^2}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d)}{c d \sqrt {c+d x^2}} \]
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Rubi [A] time = 0.27, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {413, 528, 531, 418, 492, 411} \[ -\frac {x \sqrt {a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right )}{3 c d^2 \sqrt {c+d x^2}}+\frac {\sqrt {a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-3 a d)}{3 c d^2}-\frac {2 b \sqrt {c} \sqrt {a+b x^2} (2 b c-3 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \left (a+b x^2\right )^{3/2} (b c-a d)}{c d \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 413
Rule 418
Rule 492
Rule 528
Rule 531
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2}} \, dx &=-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{c d \sqrt {c+d x^2}}+\frac {\int \frac {\sqrt {a+b x^2} \left (a b c+b (4 b c-3 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 )^{3/2}}{c d \sqrt {c+d x^2}}+\frac {b (4 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d^2}+\frac {\int \frac {-2 a b c (2 b c-3 a d)-b \left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{c d \sqrt {c+d x^2}}+\frac {b (4 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d^2}-\frac {(2 a b (2 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 d^2}-\frac {\left (b \left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c d^2}\\ &=-\frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{3 c d^2 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{c d \sqrt {c+d x^2}}+\frac {b (4 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d^2}-\frac {2 b \sqrt {c} (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 )}{3 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=-\frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{3 c d^2 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{3/2}}{c d \sqrt {c+d x^2}}+\frac {b (4 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d^2}+\frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\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 )}{3 \sqrt {c} d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {2 b \sqrt {c} (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 )}{3 d^{5/2} \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 = 0.47, size = 256, normalized size = 0.74 \[ \frac {-i b c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (9 a^2 d^2-17 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (x \sqrt {\frac {b}{a}}\right ),\frac {a d}{b c}\right )+i b c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+d x \sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (3 a^2 d^2-6 a b c d+b^2 c \left (4 c+d x^2\right )\right )}{3 c d^3 \sqrt {\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 539, normalized size = 1.56 \[ \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (\sqrt {-\frac {b}{a}}\, b^{3} c \,d^{2} x^{5}+3 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{3}-5 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{3}+4 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d \,x^{3}+3 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x -6 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} b c \,d^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+9 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} b c \,d^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+4 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d x +13 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a \,b^{2} c^{2} d \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-17 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a \,b^{2} c^{2} d \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{3} c^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{3} c^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )\right )}{3 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, c \,d^{3}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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