Optimal. Leaf size=490 \[ -\frac {b}{a (b+a c) x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(3 b-a c) \left (b+a c+a d x^2\right )}{3 a (b+a c)^2 x^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(7 b-a c) d \left (b+a c+a d x^2\right )}{3 (b+a c)^3 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(7 b-a c) d^2 x \left (b+a c+a d x^2\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} (7 b-a c) d^{3/2} \left (b+a c+a d x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {\sqrt {c} (3 b-a c) d^{3/2} \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.42, antiderivative size = 490, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986,
479, 597, 545, 429, 506, 422} \begin {gather*} \frac {\sqrt {c} d^{3/2} (3 b-a c) \left (a c+a d x^2+b\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac {\sqrt {c} d^{3/2} (7 b-a c) \left (a c+a d x^2+b\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^2 x (7 b-a c) \left (a c+a d x^2+b\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {d (7 b-a c) \left (a c+a d x^2+b\right )}{3 x (a c+b)^3 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}+\frac {(3 b-a c) \left (a c+a d x^2+b\right )}{3 a x^3 (a c+b)^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {b}{a x^3 (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 422
Rule 429
Rule 479
Rule 506
Rule 545
Rule 597
Rule 1985
Rule 1986
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (b+a \left (c+d x^2\right )\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {\left (c+d x^2\right )^{3/2}}{x^4 \left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {c (3 b-a c) d+(2 b-a c) d^2 x^2}{x^4 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{a (b+a c) d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {a c^2 (7 b-a c) d^2+a c (3 b-a c) d^3 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 a c (b+a c)^2 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {-a c^2 (3 b-a c) (b+a c) d^3-a^2 c^2 (7 b-a c) d^4 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 a c^2 (b+a c)^3 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left ((3 b-a c) d^2 \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 (b+a c)^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (a (7 b-a c) d^3 \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{3 (b+a c)^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(7 b-a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {c} (3 b-a c) d^{3/2} \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (c (7 b-a c) d^2 \sqrt {b+a \left (c+d x^2\right )}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 (b+a c)^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {b \sqrt {b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(3 b-a c) \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(7 b-a c) d \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(7 b-a c) d^2 x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {c} (7 b-a c) d^{3/2} \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {c} (3 b-a c) d^{3/2} \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 10.58, size = 319, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} \left (c+d x^2\right ) \left (b^2 \left (c+4 d x^2\right )+a^2 c \left (c^2-d^2 x^4\right )+a b \left (2 c^2+4 c d x^2+7 d^2 x^4\right )\right )-i a c (-7 b+a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )+i b (3 b-5 a c) d^2 x^3 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )\right )}{3 (b+a c)^3 \sqrt {\frac {a d}{b+a c}} x^3 \left (b+a \left (c+d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1079\) vs.
\(2(526)=1052\).
time = 0.08, size = 1080, normalized size = 2.20 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {1}{x^{4} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^4\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________