Optimal. Leaf size=192 \[ -\frac {a (7 b c-3 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 c^2 x^2}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 c x^4}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 c^{5/2}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, 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 = {457, 100, 154,
163, 65, 223, 212, 95, 214} \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.
[In]
[Out]
Rule 65
Rule 95
Rule 100
Rule 154
Rule 163
Rule 212
Rule 214
Rule 223
Rule 457
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} \text {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 {\text {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 {\text {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 \text {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 ) \text {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 \text {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 ) \text {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 \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 3.54, size = 206, normalized size = 1.07 \begin {gather*} \frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-2 a c-9 b c x^2+3 a d 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}}-\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs.
\(2(154)=308\).
time = 0.14, size = 408, normalized size = 2.12
method | result | size |
risch | \(-\frac {a \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-3 a d \,x^{2}+9 c \,x^{2} b +2 a c \right )}{8 c^{2} x^{4}}+\frac {\left (\frac {b^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{2 \sqrt {b d}}-\frac {3 a^{3} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d^{2}}{16 c^{2} \sqrt {a c}}+\frac {5 a^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b d}{8 c \sqrt {a c}}-\frac {15 a \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b^{2}}{16 \sqrt {a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(336\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{2 \sqrt {b d}}-\frac {15 a \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b^{2}}{16 \sqrt {a c}}-\frac {9 a b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{8 c \,x^{2}}+\frac {5 a^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) b d}{8 c \sqrt {a c}}-\frac {a^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{4 c \,x^{4}}+\frac {3 a^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, d}{8 c^{2} x^{2}}-\frac {3 a^{3} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d^{2}}{16 c^{2} \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(391\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (8 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{4} \sqrt {a c}-3 \ln \left (\frac {a d \,x^{2}+c \,x^{2} b +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{3} d^{2} x^{4} \sqrt {b d}+10 \ln \left (\frac {a d \,x^{2}+c \,x^{2} b +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{2} b c d \,x^{4} \sqrt {b d}-15 \ln \left (\frac {a d \,x^{2}+c \,x^{2} b +2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a \,b^{2} c^{2} x^{4} \sqrt {b d}+6 a^{2} d \,x^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a c}\, \sqrt {b d}-18 a b c \,x^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a c}\, \sqrt {b d}-4 a^{2} c \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a c}\, \sqrt {b d}\right )}{16 c^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x^{4} \sqrt {a c}\, \sqrt {b d}}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.70, size = 1123, normalized size = 5.85 \begin {gather*} \left [\frac {8 \, b^{2} c^{2} x^{4} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, c^{2} x^{4}}, -\frac {16 \, b^{2} c^{2} x^{4} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) - {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) + 4 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, c^{2} x^{4}}, \frac {4 \, b^{2} c^{2} x^{4} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) - 2 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, c^{2} x^{4}}, -\frac {8 \, b^{2} c^{2} x^{4} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) - {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) + 2 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, c^{2} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1175 vs.
\(2 (154) = 308\).
time = 1.70, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________