Optimal. Leaf size=170 \[ \frac {a}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 a d+2 b c}{2 \sqrt {c+d x^2} (b c-a d)^3}+\frac {3 a d+2 b c}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac {\sqrt {b} (3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 78, 51, 63, 208} \[ \frac {a}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 a d+2 b c}{2 \sqrt {c+d x^2} (b c-a d)^3}+\frac {3 a d+2 b c}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac {\sqrt {b} (3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2 (c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx,x,x^2\right )}{4 b (b c-a d)}\\ &=\frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{4 (b c-a d)^2}\\ &=\frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(b (2 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 (b c-a d)^3}\\ &=\frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(b (2 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 d (b c-a d)^3}\\ &=\frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {b} (2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 93, normalized size = 0.55 \[ \frac {\left (a+b x^2\right ) (3 a d+2 b c) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \left (d x^2+c\right )}{b c-a d}\right )+3 a (b c-a d)}{6 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.09, size = 993, normalized size = 5.84 \[ \left [-\frac {3 \, {\left ({\left (2 \, b^{2} c d^{2} + 3 \, a b d^{3}\right )} x^{6} + 2 \, a b c^{3} + 3 \, a^{2} c^{2} d + {\left (4 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + 7 \, a b c^{2} d + 6 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \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 (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} x^{4} + 11 \, a b c^{2} + 4 \, a^{2} c d + 2 \, {\left (4 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (2 \, b^{2} c d^{2} + 3 \, a b d^{3}\right )} x^{6} + 2 \, a b c^{3} + 3 \, a^{2} c^{2} d + {\left (4 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + 7 \, a b c^{2} d + 6 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 2 \, {\left (3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} x^{4} + 11 \, a b c^{2} + 4 \, a^{2} c d + 2 \, {\left (4 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 260, normalized size = 1.53 \[ \frac {\frac {3 \, \sqrt {d x^{2} + c} a b d^{2}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} + \frac {3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} b c d + b c^{2} d + 3 \, {\left (d x^{2} + c\right )} a d^{2} - a c d^{2}\right )}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 2400, normalized size = 14.12 \[ \text {result too large to display} \]
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 = 1.43, size = 193, normalized size = 1.14 \[ -\frac {\frac {\left (d\,x^2+c\right )\,\left (3\,a\,d+2\,b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^2}-\frac {c}{3\,\left (a\,d-b\,c\right )}+\frac {b\,{\left (d\,x^2+c\right )}^2\,\left (3\,a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^3}}{b\,{\left (d\,x^2+c\right )}^{5/2}+{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-b\,c\right )}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^{7/2}}\right )\,\left (3\,a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________