Optimal. Leaf size=330 \[ \frac {\sqrt {a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (\frac {3 a^2 d}{c}+7 a b-\frac {2 b^2 c}{d}\right )}{3 \sqrt {c+d x^2}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 c d} \]
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Rubi [A] time = 0.29, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {474, 528, 531, 418, 492, 411} \[ \frac {\sqrt {a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (\frac {3 a^2 d}{c}+7 a b-\frac {2 b^2 c}{d}\right )}{3 \sqrt {c+d x^2}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d+b c)}{3 c d} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 474
Rule 492
Rule 528
Rule 531
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^2 \sqrt {c+d x^2}} \, dx &=-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}+\frac {\int \frac {\sqrt {a+b x^2} \left (4 a b c+b (b c+3 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{c}\\ &=\frac {b (b c+3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}+\frac {\int \frac {-a b c (b c-9 a d)-b \left (2 b^2 c^2-7 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}\\ &=\frac {b (b c+3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}-\frac {(a b (b c-9 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 d}-\frac {\left (b \left (2 b^2 c^2-7 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}\\ &=\frac {\left (7 a b-\frac {2 b^2 c}{d}+\frac {3 a^2 d}{c}\right ) x \sqrt {a+b x^2}}{3 \sqrt {c+d x^2}}+\frac {b (b c+3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}-\frac {b \sqrt {c} (b c-9 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^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (2 b^2 c^2-7 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}\\ &=\frac {\left (7 a b-\frac {2 b^2 c}{d}+\frac {3 a^2 d}{c}\right ) x \sqrt {a+b x^2}}{3 \sqrt {c+d x^2}}+\frac {b (b c+3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c d}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{c x}+\frac {\left (2 b^2 c^2-7 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^{3/2} \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} (b c-9 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^{3/2} \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 = 0.48, size = 254, normalized size = 0.77 \[ \frac {-2 i b c x \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c x \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (3 a^2 d^2+7 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+d \left (-\sqrt {\frac {b}{a}}\right ) \left (a+b x^2\right ) \left (c+d x^2\right ) \left (3 a^2 d-b^2 c x^2\right )}{3 c d^2 x \sqrt {\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{d x^{4} + c x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 568, normalized size = 1.72 \[ \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (\sqrt {-\frac {b}{a}}\, b^{3} c \,d^{2} x^{6}-3 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{4}+\sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{4}+\sqrt {-\frac {b}{a}}\, b^{3} c^{2} d \,x^{4}-3 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x^{2}-3 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} b c \,d^{2} x \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} b c \,d^{2} x \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d \,x^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a \,b^{2} c^{2} d x \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a \,b^{2} c^{2} d x \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{3} c^{3} x \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{3} c^{3} x \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-3 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{2}\right )}{3 \left (x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, c \,d^{2} x} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^2\,\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x^{2} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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