Optimal. Leaf size=124 \[ \frac {d (2 b c-a d) \sqrt {c+d x^2}}{b^2}+\frac {d \left (c+d x^2\right )^{3/2}}{3 b}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}+\frac {(b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a b^{5/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 124, 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, 86, 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 b^{5/2}}+\frac {d \sqrt {c+d x^2} (2 b c-a d)}{b^2}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}+\frac {d \left (c+d x^2\right )^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 86
Rule 159
Rule 162
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{x \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac {d \left (c+d x^2\right )^{3/2}}{3 b}+\frac {\text {Subst}\left (\int \frac {\sqrt {c+d x} \left (b c^2+d (2 b c-a d) x\right )}{x (a+b x)} \, dx,x,x^2\right )}{2 b}\\ &=\frac {d (2 b c-a d) \sqrt {c+d x^2}}{b^2}+\frac {d \left (c+d x^2\right )^{3/2}}{3 b}+\frac {\text {Subst}\left (\int \frac {\frac {b^2 c^3}{2}+\frac {1}{2} d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{b^2}\\ &=\frac {d (2 b c-a d) \sqrt {c+d x^2}}{b^2}+\frac {d \left (c+d x^2\right )^{3/2}}{3 b}+\frac {c^3 \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a}-\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 b^2}\\ &=\frac {d (2 b c-a d) \sqrt {c+d x^2}}{b^2}+\frac {d \left (c+d x^2\right )^{3/2}}{3 b}+\frac {c^3 \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a 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 b^2 d}\\ &=\frac {d (2 b c-a d) \sqrt {c+d x^2}}{b^2}+\frac {d \left (c+d x^2\right )^{3/2}}{3 b}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a}+\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 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 114, normalized size = 0.92 \begin {gather*} \frac {d \sqrt {c+d x^2} \left (7 b c-3 a d+b d x^2\right )}{3 b^2}+\frac {(-b c+a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{a b^{5/2}}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a} \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. \(2137\) vs.
\(2(102)=204\).
time = 0.10, size = 2138, normalized size = 17.24
method | result | size |
default | \(\text {Expression too large to display}\) | \(2138\) |
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 = 2.99, size = 837, normalized size = 6.75 \begin {gather*} \left [\frac {6 \, b^{2} c^{\frac {5}{2}} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \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 ) + 4 \, {\left (a b d^{2} x^{2} + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {d x^{2} + c}}{12 \, a b^{2}}, \frac {12 \, b^{2} \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \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 ) + 4 \, {\left (a b d^{2} x^{2} + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {d x^{2} + c}}{12 \, a b^{2}}, \frac {3 \, b^{2} c^{\frac {5}{2}} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \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 ) + 2 \, {\left (a b d^{2} x^{2} + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {d x^{2} + c}}{6 \, a b^{2}}, \frac {6 \, b^{2} \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \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 ) + 2 \, {\left (a b d^{2} x^{2} + 7 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {d x^{2} + c}}{6 \, a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 22.29, size = 119, normalized size = 0.96 \begin {gather*} \frac {d \left (c + d x^{2}\right )^{\frac {3}{2}}}{3 b} + \frac {\sqrt {c + d x^{2}} \left (- a d^{2} + 2 b c d\right )}{b^{2}} + \frac {c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} + \frac {\left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{a b^{3} \sqrt {\frac {a d - b c}{b}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 163, normalized size = 1.31 \begin {gather*} \frac {c^{3} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a \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 b^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d + 6 \, \sqrt {d x^{2} + c} b^{2} c d - 3 \, \sqrt {d x^{2} + c} a b d^{2}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 2094, normalized size = 16.89 \begin {gather*} \frac {d\,{\left (d\,x^2+c\right )}^{3/2}}{3\,b}-\frac {d\,\sqrt {d\,x^2+c}\,\left (a\,d-2\,b\,c\right )}{b^2}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {c^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}+\frac {\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}+\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {c^5}}{a\,b^3}\right )\,\sqrt {c^5}}{2\,a}\right )\,1{}\mathrm {i}}{2\,a}+\frac {\sqrt {c^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}-\frac {\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}-\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {c^5}}{a\,b^3}\right )\,\sqrt {c^5}}{2\,a}\right )\,1{}\mathrm {i}}{2\,a}}{\frac {2\,\left (a^5\,c^3\,d^8-6\,a^4\,b\,c^4\,d^7+15\,a^3\,b^2\,c^5\,d^6-19\,a^2\,b^3\,c^6\,d^5+12\,a\,b^4\,c^7\,d^4-3\,b^5\,c^8\,d^3\right )}{b^3}-\frac {\sqrt {c^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}+\frac {\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}+\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {c^5}}{a\,b^3}\right )\,\sqrt {c^5}}{2\,a}\right )}{2\,a}+\frac {\sqrt {c^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}-\frac {\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}-\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {c^5}}{a\,b^3}\right )\,\sqrt {c^5}}{2\,a}\right )}{2\,a}}\right )\,\sqrt {c^5}\,1{}\mathrm {i}}{a}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}+\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}+\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\sqrt {d\,x^2+c}}{a\,b^8}\right )}{2\,a\,b^5}\right )\,1{}\mathrm {i}}{2\,a\,b^5}+\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}-\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}-\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\sqrt {d\,x^2+c}}{a\,b^8}\right )}{2\,a\,b^5}\right )\,1{}\mathrm {i}}{2\,a\,b^5}}{\frac {2\,\left (a^5\,c^3\,d^8-6\,a^4\,b\,c^4\,d^7+15\,a^3\,b^2\,c^5\,d^6-19\,a^2\,b^3\,c^6\,d^5+12\,a\,b^4\,c^7\,d^4-3\,b^5\,c^8\,d^3\right )}{b^3}-\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}+\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}+\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\sqrt {d\,x^2+c}}{a\,b^8}\right )}{2\,a\,b^5}\right )}{2\,a\,b^5}+\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {2\,\sqrt {d\,x^2+c}\,\left (a^6\,d^8-6\,a^5\,b\,c\,d^7+15\,a^4\,b^2\,c^2\,d^6-20\,a^3\,b^3\,c^3\,d^5+15\,a^2\,b^4\,c^4\,d^4-6\,a\,b^5\,c^5\,d^3+2\,b^6\,c^6\,d^2\right )}{b^3}-\frac {\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\left (\frac {4\,a^4\,b^3\,c\,d^5-12\,a^3\,b^4\,c^2\,d^4+8\,a^2\,b^5\,c^3\,d^3}{b^3}-\frac {\left (4\,a^3\,b^5\,d^3-8\,a^2\,b^6\,c\,d^2\right )\,\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,\sqrt {d\,x^2+c}}{a\,b^8}\right )}{2\,a\,b^5}\right )}{2\,a\,b^5}}\right )\,\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^5}\,1{}\mathrm {i}}{a\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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