Optimal. Leaf size=166 \[ \frac {a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^3}-\frac {(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 d^{3/2}}+\frac {\sqrt {a+\frac {b}{x}} (b c-2 a d) (b c-a d)}{c^2 d \left (c+\frac {d}{x}\right )}+\frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )} \]
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Rubi [A] time = 0.23, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {375, 98, 149, 156, 63, 208, 205} \[ \frac {a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^3}-\frac {(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 d^{3/2}}+\frac {\sqrt {a+\frac {b}{x}} (b c-2 a d) (b c-a d)}{c^2 d \left (c+\frac {d}{x}\right )}+\frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 149
Rule 156
Rule 205
Rule 208
Rule 375
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (5 b c-4 a d)-\frac {1}{2} b (2 b c-a d) x\right )}{x (c+d x)^2} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {(b c-2 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{c^2 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} a^2 d (5 b c-4 a d)+\frac {1}{2} b \left (b^2 c^2+2 a b c d-2 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{c^2 d}\\ &=\frac {(b c-2 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{c^2 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )}-\frac {\left (a^2 (5 b c-4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 c^3}-\frac {\left ((b c-a d)^2 (b c+4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{2 c^3 d}\\ &=\frac {(b c-2 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{c^2 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )}-\frac {\left (a^2 (5 b c-4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^3}-\frac {\left ((b c-a d)^2 (b c+4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^3 d}\\ &=\frac {(b c-2 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{c^2 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )}-\frac {(b c-a d)^{3/2} (b c+4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 d^{3/2}}+\frac {a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^3}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 145, normalized size = 0.87 \[ \frac {a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+\frac {c x \sqrt {a+\frac {b}{x}} \left (a^2 d (c x+2 d)-2 a b c d+b^2 c^2\right )}{d (c x+d)}-\frac {(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{d^{3/2}}}{c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 1001, normalized size = 6.03 \[ \left [-\frac {{\left (5 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + {\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{d}} \log \left (\frac {2 \, d x \sqrt {-\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}} + b d - {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (a^{2} c^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (c^{4} d x + c^{3} d^{2}\right )}}, -\frac {2 \, {\left (5 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{d}} \log \left (\frac {2 \, d x \sqrt {-\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}} + b d - {\left (b c - 2 \, a d\right )} x}{c x + d}\right ) - 2 \, {\left (a^{2} c^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (c^{4} d x + c^{3} d^{2}\right )}}, \frac {2 \, {\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{d}} \arctan \left (-\frac {d \sqrt {\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}}}{b c - a d}\right ) - {\left (5 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (a^{2} c^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (c^{4} d x + c^{3} d^{2}\right )}}, \frac {{\left (b^{2} c^{2} d + 3 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{d}} \arctan \left (-\frac {d \sqrt {\frac {b c - a d}{d}} \sqrt {\frac {a x + b}{x}}}{b c - a d}\right ) - {\left (5 \, a b c d^{2} - 4 \, a^{2} d^{3} + {\left (5 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} c^{2} d x^{2} + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{c^{4} d x + c^{3} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1323, normalized size = 7.97 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{{\left (c + \frac {d}{x}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.31, size = 1153, normalized size = 6.95 \[ \frac {\frac {\sqrt {a+\frac {b}{x}}\,\left (2\,a^3\,b\,d^2-3\,a^2\,b^2\,c\,d+a\,b^3\,c^2\right )}{c^2\,d}-\frac {b\,{\left (a+\frac {b}{x}\right )}^{3/2}\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{c^2\,d}}{\left (a+\frac {b}{x}\right )\,\left (2\,a\,d-b\,c\right )-d\,{\left (a+\frac {b}{x}\right )}^2-a^2\,d+a\,b\,c}-\frac {\mathrm {atanh}\left (\frac {10\,b^9\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{10\,a^2\,b^9+\frac {32\,a^3\,b^8\,d}{c}-\frac {132\,a^4\,b^7\,d^2}{c^2}+\frac {130\,a^5\,b^6\,d^3}{c^3}-\frac {40\,a^6\,b^5\,d^4}{c^4}}+\frac {32\,a\,b^8\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8+\frac {10\,a^2\,b^9\,c}{d}-\frac {132\,a^4\,b^7\,d}{c}+\frac {130\,a^5\,b^6\,d^2}{c^2}-\frac {40\,a^6\,b^5\,d^3}{c^3}}-\frac {132\,a^2\,b^7\,d\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8\,c-132\,a^4\,b^7\,d+\frac {10\,a^2\,b^9\,c^2}{d}+\frac {130\,a^5\,b^6\,d^2}{c}-\frac {40\,a^6\,b^5\,d^3}{c^2}}+\frac {130\,a^3\,b^6\,d^2\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8\,c^2+130\,a^5\,b^6\,d^2+\frac {10\,a^2\,b^9\,c^3}{d}-\frac {40\,a^6\,b^5\,d^3}{c}-132\,a^4\,b^7\,c\,d}-\frac {40\,a^4\,b^5\,d^3\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3}}{32\,a^3\,b^8\,c^3-40\,a^6\,b^5\,d^3-132\,a^4\,b^7\,c^2\,d+130\,a^5\,b^6\,c\,d^2+\frac {10\,a^2\,b^9\,c^4}{d}}\right )\,\left (4\,a\,d-5\,b\,c\right )\,\sqrt {a^3}}{c^3}+\frac {\mathrm {atanh}\left (\frac {30\,a^3\,b^6\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{14\,a^2\,b^9\,c^3+110\,a^5\,b^6\,d^3-4\,a^3\,b^8\,c^2\,d-82\,a^4\,b^7\,c\,d^2+\frac {2\,a\,b^{10}\,c^4}{d}-\frac {40\,a^6\,b^5\,d^4}{c}}+\frac {18\,a^2\,b^7\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{2\,a\,b^{10}\,c^3-82\,a^4\,b^7\,d^3+14\,a^2\,b^9\,c^2\,d-4\,a^3\,b^8\,c\,d^2+\frac {110\,a^5\,b^6\,d^4}{c}-\frac {40\,a^6\,b^5\,d^5}{c^2}}+\frac {40\,a^4\,b^5\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{4\,a^3\,b^8\,c^3+40\,a^6\,b^5\,d^3+82\,a^4\,b^7\,c^2\,d-110\,a^5\,b^6\,c\,d^2-\frac {2\,a\,b^{10}\,c^5}{d^2}-\frac {14\,a^2\,b^9\,c^4}{d}}-\frac {2\,a\,b^8\,\sqrt {a+\frac {b}{x}}\,\sqrt {a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3}}{4\,a^3\,b^8\,d^3-14\,a^2\,b^9\,c\,d^2+\frac {82\,a^4\,b^7\,d^4}{c}-\frac {110\,a^5\,b^6\,d^5}{c^2}+\frac {40\,a^6\,b^5\,d^6}{c^3}-2\,a\,b^{10}\,c^2\,d}\right )\,\sqrt {d^3\,{\left (a\,d-b\,c\right )}^3}\,\left (4\,a\,d+b\,c\right )}{c^3\,d^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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