Optimal. Leaf size=180 \[ -\frac {(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 b^{3/2}}+\frac {c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3}-\frac {\sqrt {c+d x^2} (b c-a d) (2 b c-a d)}{2 a^2 b \left (a+b x^2\right )}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.27, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 98, 149, 156, 63, 208} \[ -\frac {(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 b^{3/2}}+\frac {c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3}-\frac {\sqrt {c+d x^2} (b c-a d) (2 b c-a d)}{2 a^2 b \left (a+b x^2\right )}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 149
Rule 156
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{x^3 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^2} \, dx,x,x^2\right )\\ &=-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (4 b c-5 a d)+\frac {1}{2} d (b c-2 a d) x\right )}{x (a+b x)^2} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 b \left (a+b x^2\right )}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} b c^2 (4 b c-5 a d)-\frac {1}{2} d \left (2 b^2 c^2-2 a b c d-a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2 b}\\ &=-\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 b \left (a+b x^2\right )}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )}-\frac {\left (c^2 (4 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^3}+\frac {\left ((b c-a d)^2 (4 b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^3 b}\\ &=-\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 b \left (a+b x^2\right )}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )}-\frac {\left (c^2 (4 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^3 d}+\frac {\left ((b c-a d)^2 (4 b c+a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^3 b d}\\ &=-\frac {(b c-a d) (2 b c-a d) \sqrt {c+d x^2}}{2 a^2 b \left (a+b x^2\right )}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )}+\frac {c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3}-\frac {(b c-a d)^{3/2} (4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 175, normalized size = 0.97 \[ -\frac {\frac {a \sqrt {c+d x^2} \left (a^2 d^2 x^2+a b c \left (c-2 d x^2\right )+2 b^2 c^2 x^2\right )}{b x^2 \left (a+b x^2\right )}+\frac {\sqrt {b c-a d} \left (-a^2 d^2-3 a b c d+4 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{3/2}}-\left (c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\right )}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.35, size = 1266, normalized size = 7.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 283, normalized size = 1.57 \[ -\frac {{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c}} + \frac {{\left (4 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{3} b} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{2} d - 2 \, \sqrt {d x^{2} + c} b^{2} c^{3} d - 2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c d^{2} + 3 \, \sqrt {d x^{2} + c} a b c^{2} d^{2} + {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3} - \sqrt {d x^{2} + c} a^{2} c d^{3}}{2 \, {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 7590, normalized size = 42.17 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.99, size = 1152, normalized size = 6.40 \[ \frac {\frac {\sqrt {d\,x^2+c}\,\left (a^2\,c\,d^3-3\,a\,b\,c^2\,d^2+2\,b^2\,c^3\,d\right )}{2\,a^2\,b}-\frac {d\,{\left (d\,x^2+c\right )}^{3/2}\,\left (a^2\,d^2-2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,a^2\,b}}{\left (d\,x^2+c\right )\,\left (a\,d-2\,b\,c\right )+b\,{\left (d\,x^2+c\right )}^2+b\,c^2-a\,c\,d}-\frac {\mathrm {atanh}\left (\frac {5\,d^9\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{4\,\left (\frac {5\,c^2\,d^9}{4}+\frac {4\,b\,c^3\,d^8}{a}-\frac {33\,b^2\,c^4\,d^7}{2\,a^2}+\frac {65\,b^3\,c^5\,d^6}{4\,a^3}-\frac {5\,b^4\,c^6\,d^5}{a^4}\right )}+\frac {4\,c\,d^8\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{4\,c^3\,d^8+\frac {5\,a\,c^2\,d^9}{4\,b}-\frac {33\,b\,c^4\,d^7}{2\,a}+\frac {65\,b^2\,c^5\,d^6}{4\,a^2}-\frac {5\,b^3\,c^6\,d^5}{a^3}}+\frac {65\,b^2\,c^3\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{4\,\left (4\,a^2\,c^3\,d^8+\frac {65\,b^2\,c^5\,d^6}{4}-\frac {5\,b^3\,c^6\,d^5}{a}+\frac {5\,a^3\,c^2\,d^9}{4\,b}-\frac {33\,a\,b\,c^4\,d^7}{2}\right )}-\frac {5\,b^3\,c^4\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{4\,a^3\,c^3\,d^8-5\,b^3\,c^6\,d^5+\frac {65\,a\,b^2\,c^5\,d^6}{4}-\frac {33\,a^2\,b\,c^4\,d^7}{2}+\frac {5\,a^4\,c^2\,d^9}{4\,b}}-\frac {33\,b\,c^2\,d^7\,\sqrt {d\,x^2+c}\,\sqrt {c^3}}{2\,\left (4\,a\,c^3\,d^8-\frac {33\,b\,c^4\,d^7}{2}+\frac {65\,b^2\,c^5\,d^6}{4\,a}+\frac {5\,a^2\,c^2\,d^9}{4\,b}-\frac {5\,b^3\,c^6\,d^5}{a^2}\right )}\right )\,\left (5\,a\,d-4\,b\,c\right )\,\sqrt {c^3}}{2\,a^3}-\frac {\mathrm {atanh}\left (\frac {15\,c^3\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{4\,\left (\frac {7\,a^3\,c^2\,d^9}{4}+\frac {55\,b^3\,c^5\,d^6}{4}-\frac {41\,a\,b^2\,c^4\,d^7}{4}-\frac {a^2\,b\,c^3\,d^8}{2}+\frac {a^4\,c\,d^{10}}{4\,b}-\frac {5\,b^4\,c^6\,d^5}{a}\right )}+\frac {9\,c^2\,d^7\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{4\,\left (\frac {a^3\,c\,d^{10}}{4}-\frac {41\,b^3\,c^4\,d^7}{4}-\frac {a\,b^2\,c^3\,d^8}{2}+\frac {7\,a^2\,b\,c^2\,d^9}{4}+\frac {55\,b^4\,c^5\,d^6}{4\,a}-\frac {5\,b^5\,c^6\,d^5}{a^2}\right )}+\frac {5\,c^4\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{\frac {a^3\,c^3\,d^8}{2}+5\,b^3\,c^6\,d^5-\frac {55\,a\,b^2\,c^5\,d^6}{4}+\frac {41\,a^2\,b\,c^4\,d^7}{4}-\frac {a^5\,c\,d^{10}}{4\,b^2}-\frac {7\,a^4\,c^2\,d^9}{4\,b}}-\frac {c\,d^8\,\sqrt {d\,x^2+c}\,\sqrt {-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}}{4\,\left (\frac {b^3\,c^3\,d^8}{2}-\frac {7\,a\,b^2\,c^2\,d^9}{4}+\frac {41\,b^4\,c^4\,d^7}{4\,a}-\frac {55\,b^5\,c^5\,d^6}{4\,a^2}+\frac {5\,b^6\,c^6\,d^5}{a^3}-\frac {a^2\,b\,c\,d^{10}}{4}\right )}\right )\,\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^3}\,\left (a\,d+4\,b\,c\right )}{2\,a^3\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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