Optimal. Leaf size=447 \[ \frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {d^{3/4} (14 b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 c^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.63, antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 482, 541,
537, 230, 227, 418, 1233, 1232} \begin {gather*} \frac {d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 c^{3/4} \sqrt {c-d x^2} (b c-a d)^3}-\frac {b \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (9 a d+b c) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}-\frac {b \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (9 a d+b c) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d e \sqrt {e x} (a d+14 b c)}{6 c \sqrt {c-d x^2} (b c-a d)^3}+\frac {5 d e \sqrt {e x}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac {e \sqrt {e x}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 230
Rule 418
Rule 477
Rule 482
Rule 537
Rule 541
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^4}{\left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {e \text {Subst}\left (\int \frac {c+\frac {9 d x^4}{e^2}}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)}\\ &=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {e^3 \text {Subst}\left (\int \frac {-\frac {2 c (3 b c+2 a d)}{e^2}-\frac {50 b c d x^4}{e^4}}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{12 c (b c-a d)^2}\\ &=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {e^5 \text {Subst}\left (\int \frac {\frac {4 c \left (3 b^2 c^2+13 a b c d-a^2 d^2\right )}{e^4}+\frac {4 b c d (14 b c+a d) x^4}{e^6}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{24 c^2 (b c-a d)^3}\\ &=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {(d (14 b c+a d) e) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 c (b c-a d)^3}-\frac {(b (b c+9 a d) e) \text {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^3}\\ &=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {(b (b c+9 a d) e) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3}-\frac {(b (b c+9 a d) e) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3}+\frac {\left (d (14 b c+a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{6 c (b c-a d)^3 \sqrt {c-d x^2}}\\ &=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {d^{3/4} (14 b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 c^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (b (b c+9 a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (b (b c+9 a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3 \sqrt {c-d x^2}}\\ &=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {d^{3/4} (14 b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 c^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.36, size = 275, normalized size = 0.62 \begin {gather*} \frac {e \sqrt {e x} \left (5 a \left (a^2 d^2 \left (c+d x^2\right )+b^2 c \left (-3 c^2+19 c d x^2-14 d^2 x^4\right )-a b d \left (13 c^2-10 c d x^2+d^2 x^4\right )\right )-5 \left (-3 b^2 c^2-13 a b c d+a^2 d^2\right ) \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d (14 b c+a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a c (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4390\) vs.
\(2(353)=706\).
time = 0.14, size = 4391, normalized size = 9.82
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {b^{2} d e \sqrt {-d e \,x^{3}+c e x}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right ) \left (b d \,x^{2}-a d \right )}+\frac {e \sqrt {-d e \,x^{3}+c e x}}{3 \left (a d -b c \right )^{2} d \left (x^{2}-\frac {c}{d}\right )^{2}}-\frac {d \,e^{2} x \left (a d +11 b c \right )}{6 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}-\frac {7 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b \,e^{2}}{6 \sqrt {-d e \,x^{3}+c e x}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right )}-\frac {d \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{2} a}{12 \sqrt {-d e \,x^{3}+c e x}\, c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right )}-\frac {9 e^{2} b \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right ) \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e^{2} b^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right ) \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {9 e^{2} b \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right ) \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {e^{2} b^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right ) \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-d \,x^{2}+c}}\) | \(1192\) |
default | \(\text {Expression too large to display}\) | \(4391\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (e x\right )^{\frac {3}{2}}}{\left (- a + b x^{2}\right )^{2} \left (c - d x^{2}\right )^{\frac {5}{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 {{\left (e\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________