Optimal. Leaf size=154 \[ -\frac {\sqrt {c+d x^2} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right )}{15 b^2 \sqrt {a+b x^2} (b c-a d)^3}-\frac {a^2 \sqrt {c+d x^2}}{5 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac {2 a \sqrt {c+d x^2} (5 b c-3 a d)}{15 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {446, 89, 78, 37} \[ -\frac {\sqrt {c+d x^2} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right )}{15 b^2 \sqrt {a+b x^2} (b c-a d)^3}-\frac {a^2 \sqrt {c+d x^2}}{5 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)}+\frac {2 a \sqrt {c+d x^2} (5 b c-3 a d)}{15 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 78
Rule 89
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=-\frac {a^2 \sqrt {c+d x^2}}{5 b^2 (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} a (5 b c-a d)+\frac {5}{2} b (b c-a d) x}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{5 b^2 (b c-a d)}\\ &=-\frac {a^2 \sqrt {c+d x^2}}{5 b^2 (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {2 a (5 b c-3 a d) \sqrt {c+d x^2}}{15 b^2 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {\left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{30 b^2 (b c-a d)^2}\\ &=-\frac {a^2 \sqrt {c+d x^2}}{5 b^2 (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {2 a (5 b c-3 a d) \sqrt {c+d x^2}}{15 b^2 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}-\frac {\left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{15 b^2 (b c-a d)^3 \sqrt {a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 91, normalized size = 0.59 \[ -\frac {\sqrt {c+d x^2} \left (a^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )+10 a b c x^2 \left (2 c-d x^2\right )+15 b^2 c^2 x^4\right )}{15 \left (a+b x^2\right )^{5/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.15, size = 262, normalized size = 1.70 \[ -\frac {{\left ({\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 4 \, {\left (5 \, a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{4} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.87, size = 597, normalized size = 3.88 \[ -\frac {2 \, {\left (15 \, \sqrt {b d} b^{8} c^{4} - 40 \, \sqrt {b d} a b^{7} c^{3} d + 38 \, \sqrt {b d} a^{2} b^{6} c^{2} d^{2} - 16 \, \sqrt {b d} a^{3} b^{5} c d^{3} + 3 \, \sqrt {b d} a^{4} b^{4} d^{4} - 60 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{6} c^{3} + 80 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{2} d - 20 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c d^{2} + 90 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} - 40 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{3} c d + 30 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{2} - 60 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} b^{2} c + 15 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{8}\right )}}{15 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 119, normalized size = 0.77 \[ \frac {\sqrt {d \,x^{2}+c}\, \left (3 a^{2} d^{2} x^{4}-10 a b c d \,x^{4}+15 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}+20 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right )}{15 \left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (a^{3} d^{3}-3 a^{2} c \,d^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.60, size = 220, normalized size = 1.43 \[ \frac {\sqrt {b\,x^2+a}\,\left (\frac {8\,a^2\,c^3}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x^4\,\left (-a^2\,c\,d^2+10\,a\,b\,c^2\,d+15\,b^2\,c^3\right )}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x^6\,\left (3\,a^2\,d^3-10\,a\,b\,c\,d^2+15\,b^2\,c^2\,d\right )}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,a\,c^2\,x^2\,\left (a\,d+5\,b\,c\right )}{15\,b^3\,{\left (a\,d-b\,c\right )}^3}\right )}{x^6\,\sqrt {d\,x^2+c}+\frac {a^3\,\sqrt {d\,x^2+c}}{b^3}+\frac {3\,a\,x^4\,\sqrt {d\,x^2+c}}{b}+\frac {3\,a^2\,x^2\,\sqrt {d\,x^2+c}}{b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (a + b x^{2}\right )^{\frac {7}{2}} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________