Optimal. Leaf size=192 \[ -\frac {\sqrt {a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-3 a d)}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4} \]
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Rubi [A] time = 0.20, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {446, 98, 149, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {\sqrt {a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-3 a d)}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 98
Rule 149
Rule 157
Rule 206
Rule 208
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (7 b c-3 a d)-2 b^2 c x\right )}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{4} a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )-2 b^3 c^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 c^2}\\ &=-\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}+\frac {1}{2} b^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )+\frac {\left (a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{16 c^2}\\ &=-\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}+b^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )+\frac {\left (a \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{8 c^2}\\ &=-\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+b^2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )\\ &=-\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 206, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a c+3 a d x^2-9 b c x^2\right )}{8 c^2 x^4}+\frac {(b c-a d)^{5/2} \left (\frac {b \left (c+d x^2\right )}{b c-a d}\right )^{5/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \left (c+d x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^5 \sqrt {c+d x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 6.71, size = 1123, normalized size = 5.85
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.18, size = 1175, normalized size = 6.12 \begin {gather*} -\frac {{\left (\frac {4 \, \sqrt {b d} b^{2} \log \left ({\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{d} + \frac {{\left (15 \, \sqrt {b d} a b^{3} c^{2} - 10 \, \sqrt {b d} a^{2} b^{2} c d + 3 \, \sqrt {b d} a^{3} b d^{2}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{2}} + \frac {2 \, {\left (9 \, \sqrt {b d} a b^{9} c^{5} - 39 \, \sqrt {b d} a^{2} b^{8} c^{4} d + 66 \, \sqrt {b d} a^{3} b^{7} c^{3} d^{2} - 54 \, \sqrt {b d} a^{4} b^{6} c^{2} d^{3} + 21 \, \sqrt {b d} a^{5} b^{5} c d^{4} - 3 \, \sqrt {b d} a^{6} b^{4} d^{5} - 27 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{4} + 40 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c^{3} d + 10 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} c^{2} d^{2} - 32 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{4} b^{4} c d^{3} + 9 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{5} b^{3} d^{4} + 27 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{5} c^{3} + 9 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} c^{2} d + 21 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{3} b^{3} c d^{2} - 9 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{4} b^{2} d^{3} - 9 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a b^{3} c^{2} - 10 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a^{2} b^{2} c d + 3 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a^{3} b d^{2}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )}^{2} c^{2}}\right )} b}{8 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 464, normalized size = 2.42 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-3 \sqrt {b d}\, a^{3} d^{2} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+10 \sqrt {b d}\, a^{2} b c d \,x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-15 \sqrt {b d}\, a \,b^{2} c^{2} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+8 \sqrt {a c}\, b^{3} c^{2} x^{4} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, a^{2} d \,x^{2}-18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, a b c \,x^{2}-4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, a^{2} c \right )}{16 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, \sqrt {a c}\, c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^5\,\sqrt {d\,x^2+c}} \,d x \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 {5}{2}}}{x^{5} \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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