Optimal. Leaf size=918 \[ -\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 e \sqrt {c x^2+b x+a} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt {d+e x}}+\frac {8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \left (5 a c e (2 c d-b e)^2-4 c \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}} \]
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Rubi [A] time = 1.46, antiderivative size = 918, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {740, 822, 834, 843, 718, 424, 419} \[ -\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 e \sqrt {c x^2+b x+a} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt {d+e x}}+\frac {8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \left (5 a c e (2 c d-b e)^2-4 c \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 740
Rule 822
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-3 b c d e-4 b^2 e^2+14 a c e^2\right )+\frac {5}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} e \left (3 b^3 c d e^2-8 b^4 e^3+12 a c^2 e \left (c d^2-7 a e^2\right )-4 b c^2 d \left (2 c d^2+9 a e^2\right )+3 b^2 c e \left (3 c d^2+19 a e^2\right )\right )+c e (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {1}{8} c e \left (4 b^4 d e^3+3 b^2 c d e \left (5 c d^2-11 a e^2\right )+4 a c^2 d e \left (c d^2+33 a e^2\right )-b^3 \left (3 c d^2 e^2-4 a e^4\right )-4 b c \left (2 c^2 d^4+9 a c d^2 e^2+6 a^2 e^4\right )\right )+\frac {1}{8} c e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (c \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (\sqrt {2} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}+\frac {\left (8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 13.84, size = 7870, normalized size = 8.57 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{3} e^{2} x^{8} + {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{7} + {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{6} + {\left (3 \, b c^{2} d^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e + {\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{5} + a^{3} d^{2} + {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{4} + {\left (3 \, a^{2} b e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} + 6 \, {\left (a b^{2} + a^{2} c\right )} d e\right )} x^{3} + {\left (6 \, a^{2} b d e + a^{3} e^{2} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{2} + {\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 27157, normalized size = 29.58 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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