Optimal. Leaf size=144 \[ \frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}+\frac {c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2}-\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 144, 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 = {457, 100, 159,
162, 65, 214} \begin {gather*} -\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}}+\frac {c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2}+\frac {d \sqrt {c+d x^2} (2 a d+b c)}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 159
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{x^3 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\text {Subst}\left (\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (2 b c-5 a d)-\frac {1}{2} d (b c+2 a d) x\right )}{x (a+b x)} \, dx,x,x^2\right )}{2 a}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\text {Subst}\left (\int \frac {\frac {1}{4} b c^2 (2 b c-5 a d)+\frac {1}{4} d \left (b^2 c^2-6 a b c d+2 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{a b}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\left (c^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2 b}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}-\frac {\left (c^2 (2 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 d}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 b d}\\ &=\frac {d (b c+2 a d) \sqrt {c+d x^2}}{2 a b}-\frac {c \left (c+d x^2\right )^{3/2}}{2 a x^2}+\frac {c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2}-\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 125, normalized size = 0.87 \begin {gather*} \frac {\frac {a \sqrt {c+d x^2} \left (-b c^2+2 a d^2 x^2\right )}{b x^2}-\frac {2 (-b c+a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}+c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2235\) vs.
\(2(118)=236\).
time = 0.14, size = 2236, normalized size = 15.53
method | result | size |
risch | \(-\frac {c^{2} \sqrt {d \,x^{2}+c}}{2 a \,x^{2}}+\frac {d^{2} \sqrt {d \,x^{2}+c}}{b}-\frac {5 c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) d}{2 a}+\frac {c^{\frac {5}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) b}{a^{2}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) d^{3}}{2 b^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) c \,d^{2}}{2 b \sqrt {-\frac {a d -b c}{b}}}+\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) c^{2} d}{2 a \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) c^{3}}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {a \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) d^{3}}{2 b^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) c \,d^{2}}{2 b \sqrt {-\frac {a d -b c}{b}}}+\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) c^{2} d}{2 a \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) c^{3}}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}\) | \(1325\) |
default | \(\text {Expression too large to display}\) | \(2236\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.24, size = 891, normalized size = 6.19 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{4 \, a^{2} b x^{2}}, -\frac {2 \, {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{4 \, a^{2} b x^{2}}, -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{4 \, a^{2} b x^{2}}, -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, b^{2} c^{2} - 5 \, a b c d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, a^{2} d^{2} x^{2} - a b c^{2}\right )} \sqrt {d x^{2} + c}}{2 \, a^{2} b x^{2}}\right ] \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 (c + d x^{2}\right )^{\frac {5}{2}}}{x^{3} \left (a + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 158, normalized size = 1.10 \begin {gather*} \frac {\sqrt {d x^{2} + c} d^{2}}{b} - \frac {\sqrt {d x^{2} + c} c^{2}}{2 \, a x^{2}} - \frac {{\left (2 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, 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 )}{\sqrt {-b^{2} c + a b d} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.74, size = 1428, normalized size = 9.92 \begin {gather*} \frac {d^2\,\sqrt {d\,x^2+c}}{b}-\frac {b\,c^2\,d\,\sqrt {d\,x^2+c}}{2\,a\,\left (b\,\left (d\,x^2+c\right )-b\,c\right )}+\frac {\mathrm {atan}\left (\frac {a^3\,d^9\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,5{}\mathrm {i}}{5\,a^3\,c^2\,d^9-\frac {395\,b^3\,c^5\,d^6}{4}+87\,a\,b^2\,c^4\,d^7-32\,a^2\,b\,c^3\,d^8+\frac {185\,b^4\,c^6\,d^5}{4\,a}-\frac {15\,b^5\,c^7\,d^4}{2\,a^2}}+\frac {a^2\,c\,d^8\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,32{}\mathrm {i}}{32\,a^2\,c^3\,d^8+\frac {395\,b^2\,c^5\,d^6}{4}-\frac {185\,b^3\,c^6\,d^5}{4\,a}-\frac {5\,a^3\,c^2\,d^9}{b}+\frac {15\,b^4\,c^7\,d^4}{2\,a^2}-87\,a\,b\,c^4\,d^7}+\frac {b^2\,c^3\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,395{}\mathrm {i}}{4\,\left (32\,a^2\,c^3\,d^8+\frac {395\,b^2\,c^5\,d^6}{4}-\frac {185\,b^3\,c^6\,d^5}{4\,a}-\frac {5\,a^3\,c^2\,d^9}{b}+\frac {15\,b^4\,c^7\,d^4}{2\,a^2}-87\,a\,b\,c^4\,d^7\right )}-\frac {b^3\,c^4\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,185{}\mathrm {i}}{4\,\left (32\,a^3\,c^3\,d^8-\frac {185\,b^3\,c^6\,d^5}{4}+\frac {395\,a\,b^2\,c^5\,d^6}{4}-87\,a^2\,b\,c^4\,d^7+\frac {15\,b^4\,c^7\,d^4}{2\,a}-\frac {5\,a^4\,c^2\,d^9}{b}\right )}+\frac {b^4\,c^5\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,15{}\mathrm {i}}{2\,\left (32\,a^4\,c^3\,d^8+\frac {15\,b^4\,c^7\,d^4}{2}-\frac {185\,a\,b^3\,c^6\,d^5}{4}-87\,a^3\,b\,c^4\,d^7+\frac {395\,a^2\,b^2\,c^5\,d^6}{4}-\frac {5\,a^5\,c^2\,d^9}{b}\right )}-\frac {a\,b\,c^2\,d^7\,\sqrt {d\,x^2+c}\,\sqrt {c^3}\,87{}\mathrm {i}}{32\,a^2\,c^3\,d^8+\frac {395\,b^2\,c^5\,d^6}{4}-\frac {185\,b^3\,c^6\,d^5}{4\,a}-\frac {5\,a^3\,c^2\,d^9}{b}+\frac {15\,b^4\,c^7\,d^4}{2\,a^2}-87\,a\,b\,c^4\,d^7}\right )\,\left (5\,a\,d-2\,b\,c\right )\,\sqrt {c^3}\,1{}\mathrm {i}}{2\,a^2}-\frac {\mathrm {atan}\left (\frac {c^3\,d^5\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,20{}\mathrm {i}}{\frac {185\,a\,b^3\,c^5\,d^6}{2}-\frac {85\,b^4\,c^6\,d^5}{2}-16\,a^4\,c^2\,d^9+56\,a^3\,b\,c^3\,d^8+\frac {2\,a^5\,c\,d^{10}}{b}-\frac {199\,a^2\,b^2\,c^4\,d^7}{2}+\frac {15\,b^5\,c^7\,d^4}{2\,a}}-\frac {c^2\,d^6\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,10{}\mathrm {i}}{2\,a^4\,c\,d^{10}+\frac {185\,b^4\,c^5\,d^6}{2}-\frac {199\,a\,b^3\,c^4\,d^7}{2}-16\,a^3\,b\,c^2\,d^9+56\,a^2\,b^2\,c^3\,d^8-\frac {85\,b^5\,c^6\,d^5}{2\,a}+\frac {15\,b^6\,c^7\,d^4}{2\,a^2}}-\frac {c^4\,d^4\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,15{}\mathrm {i}}{2\,\left (56\,a^4\,c^3\,d^8+\frac {15\,b^4\,c^7\,d^4}{2}-\frac {85\,a\,b^3\,c^6\,d^5}{2}-\frac {199\,a^3\,b\,c^4\,d^7}{2}+\frac {2\,a^6\,c\,d^{10}}{b^2}+\frac {185\,a^2\,b^2\,c^5\,d^6}{2}-\frac {16\,a^5\,c^2\,d^9}{b}\right )}+\frac {a\,c\,d^7\,\sqrt {d\,x^2+c}\,\sqrt {-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}\,2{}\mathrm {i}}{\frac {185\,b^5\,c^5\,d^6}{2}-\frac {199\,a\,b^4\,c^4\,d^7}{2}+56\,a^2\,b^3\,c^3\,d^8-16\,a^3\,b^2\,c^2\,d^9-\frac {85\,b^6\,c^6\,d^5}{2\,a}+\frac {15\,b^7\,c^7\,d^4}{2\,a^2}+2\,a^4\,b\,c\,d^{10}}\right )\,\sqrt {-b^3\,{\left (a\,d-b\,c\right )}^5}\,1{}\mathrm {i}}{a^2\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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