Optimal. Leaf size=482 \[ -\frac {x^3 \left (c+d x^2\right )}{a d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(8 b+a c) x \left (b+a c+a d x^2\right )}{5 a^3 d^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {6 x^3 \left (b+a c+a d x^2\right )}{5 a^2 d \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) x \left (b+a c+a d x^2\right )}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} \left (16 b^2+16 a b c+a^2 c^2\right ) \left (b+a c+a d x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a^4 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {c^{3/2} (8 b+a c) \left (b+a c+a d x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.44, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1985, 1986,
478, 595, 596, 545, 429, 506, 422} \begin {gather*} \frac {c^{3/2} (a c+8 b) \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 )}{5 a^3 d^{5/2} \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 {x (a c+8 b) \left (a c+a d x^2+b\right )}{5 a^3 d^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}+\frac {6 x^3 \left (a c+a d x^2+b\right )}{5 a^2 d \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {\sqrt {c} \left (a^2 c^2+16 a b c+16 b^2\right ) \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 )}{5 a^4 d^{5/2} \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 {x \left (a^2 c^2+16 a b c+16 b^2\right ) \left (a c+a d x^2+b\right )}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}-\frac {x^3 \left (c+d x^2\right )}{a d \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 478
Rule 506
Rule 545
Rule 595
Rule 596
Rule 1985
Rule 1986
Rubi steps
\begin {align*} \int \frac {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 {x^4 \left (c+d x^2\right )^{3/2}}{\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 {x^4 \left (c+d x^2\right )^{3/2}}{\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 {x^3 \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \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 {x^2 \sqrt {c+d x^2} \left (3 c+6 d x^2\right )}{\sqrt {b+a c+a d x^2}} \, dx}{a d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {6 x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^2 \left (-3 c (6 b+a c) d-3 (8 b+a c) d^2 x^2\right )}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 a^2 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(8 b+a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {6 x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {-3 c (b+a c) (8 b+a c) d^2-3 \left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 a^3 d^4 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(8 b+a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {6 x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c (b+a c) (8 b+a c) \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}{5 a^3 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (\left (16 b^2+16 a b c+a^2 c^2\right ) \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}{5 a^3 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(8 b+a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {6 x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}+\frac {c^{3/2} (8 b+a c) \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 )}{5 a^3 d^{5/2} \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 \left (16 b^2+16 a b c+a^2 c^2\right ) \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}{5 a^4 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {x^3 \left (c+d x^2\right ) \sqrt {b+a \left (c+d x^2\right )}}{a d \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(8 b+a c) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {6 x^3 \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) x \sqrt {b+a c+a d x^2} \sqrt {b+a \left (c+d x^2\right )}}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {c} \left (16 b^2+16 a b c+a^2 c^2\right ) \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 )}{5 a^4 d^{5/2} \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 {c^{3/2} (8 b+a c) \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 )}{5 a^3 d^{5/2} \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 = 296, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} x \left (c+d x^2\right ) \left (8 b^2+a b \left (9 c+2 d x^2\right )+a^2 \left (c^2-d^2 x^4\right )\right )+i c \left (16 b^2+16 a b c+a^2 c^2\right ) \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 c (8 b+7 a c) \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 )}{5 a^3 d^2 \sqrt {\frac {a d}{b+a c}} \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. \(1158\) vs.
\(2(518)=1036\).
time = 0.11, size = 1159, normalized size = 2.40 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 {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 {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]
________________________________________________________________________________________