Optimal. Leaf size=666 \[ -\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}+\frac {4 \sqrt {-a} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {f+g x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 1.04, antiderivative size = 666, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {935, 1668, 858,
733, 435, 430} \begin {gather*} \frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 e \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac {4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac {4 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 733
Rule 858
Rule 935
Rule 1668
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}-\frac {\int \frac {(d+e x)^2 \left (2 a (3 e f-4 d g)+2 (c d f-a e g) x+2 c (4 e f-3 d g) x^2\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{9 g}\\ &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {2 \int \frac {-a c g^2 \left (20 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+28 d^3 g^3\right )-c g \left (a e g^2 \left (40 e^2 f^2-72 d e f g+63 d^2 g^2\right )+c \left (8 e^3 f^4-6 d e^2 f^3 g-7 d^3 f g^3\right )\right ) x+c g^2 \left (a e^2 g^2 (e f-27 d g)-c \left (44 e^3 f^3-33 d e^2 f^2 g-42 d^2 e f g^2+21 d^3 g^3\right )\right ) x^2-c e g^3 \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{63 c g^5}\\ &=\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {4 \int \frac {\frac {1}{2} a c g^5 \left (21 a e^3 f g^2+c \left (92 e^3 f^3-258 d e^2 f^2 g+231 d^2 e f g^2-140 d^3 g^3\right )\right )+\frac {1}{2} c g^4 \left (21 a^2 e^3 g^4+3 a c e g^2 \left (2 e^2 f^2+9 d e f g-63 d^2 g^2\right )+c^2 f \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )\right ) x+\frac {3}{2} c^2 g^5 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{315 c^2 g^8}\\ &=-\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {8 \int \frac {\frac {3}{4} a c^2 g^7 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )+\frac {3}{4} c^2 g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{945 c^3 g^{10}}\\ &=-\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {\left (2 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{315 c g^5}-\frac {\left (8 \left (\frac {3}{4} a c^2 g^8 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-\frac {3}{4} c^2 f g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{945 c^3 g^{11}}\\ &=-\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {\left (4 a \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{315 \sqrt {-a} c^{3/2} g^5 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (16 a \left (\frac {3}{4} a c^2 g^8 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-\frac {3}{4} c^2 f g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{945 \sqrt {-a} c^{7/2} g^{11} \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}+\frac {4 \sqrt {-a} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (64 c e^3 f^3-216 c d e^2 f^2 g+252 c d^2 e f g^2-18 a e^3 f g^2-105 c d^3 g^3+45 a d e^2 g^3\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 25.14, size = 864, normalized size = 1.30 \begin {gather*} \frac {2 \sqrt {f+g x} \left (-c g^2 \left (a+c x^2\right ) \left (-2 a e^2 g^2 (-11 e f+45 d g+7 e g x)+c \left (-105 d^3 g^3+63 d^2 e g^2 (4 f-3 g x)-27 d e^2 g \left (8 f^2-6 f g x+5 g^2 x^2\right )+e^3 \left (64 f^3-48 f^2 g x+40 f g^2 x^2-35 g^3 x^3\right )\right )\right )-\frac {2 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )+c^2 f \left (-64 e^3 f^3+216 d e^2 f^2 g-252 d^2 e f g^2+105 d^3 g^3\right )\right ) \left (a+c x^2\right )-\sqrt {c} \left (i \sqrt {c} f-\sqrt {a} g\right ) \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )+c^2 f \left (-64 e^3 f^3+216 d e^2 f^2 g-252 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 i a^{3/2} e^3 g^3-9 a \sqrt {c} e^2 g^2 (2 e f-5 d g)-3 i \sqrt {a} c e g \left (16 e^2 f^2-54 d e f g+63 d^2 g^2\right )+c^{3/2} \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{315 c^2 g^6 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5078\) vs.
\(2(582)=1164\).
time = 0.12, size = 5079, normalized size = 7.63
method | result | size |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{3} x^{3} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{9 g}+\frac {2 \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right ) x^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{7 c g}+\frac {2 \left (\frac {2 a \,e^{3}}{9}+3 c \,d^{2} e -\frac {6 f \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 g}\right ) x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{5 c g}+\frac {2 \left (3 d \,e^{2} a +c \,d^{3}-\frac {4 f \left (\frac {2 a \,e^{3}}{9}+3 c \,d^{2} e -\frac {6 f \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 g}\right )}{5 g}-\frac {5 a \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 c}-\frac {2 f a \,e^{3}}{3 g}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 c g}+\frac {2 \left (d^{3} a -\frac {2 f a \left (\frac {2 a \,e^{3}}{9}+3 c \,d^{2} e -\frac {6 f \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 g}\right )}{5 c g}-\frac {a \left (3 d \,e^{2} a +c \,d^{3}-\frac {4 f \left (\frac {2 a \,e^{3}}{9}+3 c \,d^{2} e -\frac {6 f \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 g}\right )}{5 g}-\frac {5 a \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 c}-\frac {2 f a \,e^{3}}{3 g}\right )}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (3 a \,d^{2} e -\frac {4 f a \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 c g}-\frac {3 a \left (\frac {2 a \,e^{3}}{9}+3 c \,d^{2} e -\frac {6 f \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 g}\right )}{5 c}-\frac {2 f \left (3 d \,e^{2} a +c \,d^{3}-\frac {4 f \left (\frac {2 a \,e^{3}}{9}+3 c \,d^{2} e -\frac {6 f \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 g}\right )}{5 g}-\frac {5 a \left (3 d \,e^{2} c -\frac {8 c f \,e^{3}}{9 g}\right )}{7 c}-\frac {2 f a \,e^{3}}{3 g}\right )}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(1156\) |
risch | \(\text {Expression too large to display}\) | \(1937\) |
default | \(\text {Expression too large to display}\) | \(5079\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.60, size = 568, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (2 \, {\left (105 \, c^{2} d^{3} f^{2} g^{3} + 315 \, a c d^{3} g^{5} - 2 \, {\left (32 \, c^{2} f^{5} + 39 \, a c f^{3} g^{2} - 6 \, a^{2} f g^{4}\right )} e^{3} + 9 \, {\left (24 \, c^{2} d f^{4} g + 31 \, a c d f^{2} g^{3} - 15 \, a^{2} d g^{5}\right )} e^{2} - 126 \, {\left (2 \, c^{2} d^{2} f^{3} g^{2} + 3 \, a c d^{2} f g^{4}\right )} e\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (105 \, c^{2} d^{3} f g^{4} - {\left (64 \, c^{2} f^{4} g + 30 \, a c f^{2} g^{3} - 21 \, a^{2} g^{5}\right )} e^{3} + 9 \, {\left (24 \, c^{2} d f^{3} g^{2} + 13 \, a c d f g^{4}\right )} e^{2} - 63 \, {\left (4 \, c^{2} d^{2} f^{2} g^{3} + 3 \, a c d^{2} g^{5}\right )} e\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (105 \, c^{2} d^{3} g^{5} + {\left (35 \, c^{2} g^{5} x^{3} - 40 \, c^{2} f g^{4} x^{2} - 64 \, c^{2} f^{3} g^{2} - 22 \, a c f g^{4} + 2 \, {\left (24 \, c^{2} f^{2} g^{3} + 7 \, a c g^{5}\right )} x\right )} e^{3} + 9 \, {\left (15 \, c^{2} d g^{5} x^{2} - 18 \, c^{2} d f g^{4} x + 24 \, c^{2} d f^{2} g^{3} + 10 \, a c d g^{5}\right )} e^{2} + 63 \, {\left (3 \, c^{2} d^{2} g^{5} x - 4 \, c^{2} d^{2} f g^{4}\right )} e\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{945 \, c^{2} g^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + c x^{2}} \left (d + e x\right )^{3}}{\sqrt {f + g x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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