Optimal. Leaf size=666 \[ \frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 e \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac {4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac {4 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g} \]
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Rubi [A] time = 1.57, 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 = {921, 1654, 844, 719, 424, 419} \[ \frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 e \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac {4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (168 d^2 e f g^2-35 d^3 g^3-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac {4 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 844
Rule 921
Rule 1654
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 ) \operatorname {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 ) \operatorname {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] time = 7.88, size = 864, normalized size = 1.30 \[ \frac {2 \sqrt {f+g x} \left (-c \left (c x^2+a\right ) \left (c \left (\left (64 f^3-48 g x f^2+40 g^2 x^2 f-35 g^3 x^3\right ) e^3-27 d g \left (8 f^2-6 g x f+5 g^2 x^2\right ) e^2+63 d^2 g^2 (4 f-3 g x) e-105 d^3 g^3\right )-2 a e^2 g^2 (-11 e f+45 d g+7 e g x)\right ) g^2-\frac {2 \left (\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2+c^2 f \left (-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right )\right ) \left (c x^2+a\right ) g^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 i a^{3/2} e^3 g^3-9 a \sqrt {c} e^2 (2 e f-5 d g) g^2-3 i \sqrt {a} c e \left (16 e^2 f^2-54 d e g f+63 d^2 g^2\right ) g+c^{3/2} \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right ) g-\sqrt {c} \left (i \sqrt {c} f-\sqrt {a} g\right ) \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2+c^2 f \left (-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{315 c^2 g^6 \sqrt {c x^2+a}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {c x^{2} + a}}{\sqrt {g x + f}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 5079, normalized size = 7.63 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}} \,d x \]
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 \[ \int \frac {\sqrt {a + c x^{2}} \left (d + e x\right )^{3}}{\sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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