Optimal. Leaf size=657 \[ \frac {\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (a \left (d h^2-7 e g h+f g^2\right )+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (7 d h^2+5 e g h+7 f g^2\right )+24 c^2 d g^2\right )}{192 (g+h x)^4 \left (a h^2-b g h+c g^2\right )^3}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (a \left (d h^2-7 e g h+f g^2\right )+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (7 d h^2+5 e g h+7 f g^2\right )+24 c^2 d g^2\right )}{512 (g+h x)^2 \left (a h^2-b g h+c g^2\right )^4}+\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right ) \left (24 a^2 f h^2-4 c \left (a \left (d h^2-7 e g h+f g^2\right )+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (7 d h^2+5 e g h+7 f g^2\right )+24 c^2 d g^2\right )}{1024 \left (a h^2-b g h+c g^2\right )^{9/2}}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (h \left (12 a h (2 f g-e h)-b \left (-7 d h^2-5 e g h+17 f g^2\right )\right )+2 c g \left (h (e g-7 d h)+5 f g^2\right )\right )}{60 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.22, antiderivative size = 660, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1650, 806, 720, 724, 206} \[ \frac {\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right )}{192 (g+h x)^4 \left (a h^2-b g h+c g^2\right )^3}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g) \left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right )}{512 (g+h x)^2 \left (a h^2-b g h+c g^2\right )^4}+\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right ) \left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right )}{1024 \left (a h^2-b g h+c g^2\right )^{9/2}}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c \left (g h (e g-7 d h)+5 f g^3\right )-h \left (-12 a h (2 f g-e h)-b h (7 d h+5 e g)+17 b f g^2\right )\right )}{60 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )^2}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 720
Rule 724
Rule 806
Rule 1650
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}-\frac {\int \frac {\left (\frac {1}{2} \left (-12 c d g+5 b e g+12 a f g-\frac {5 b f g^2}{h}+7 b d h-12 a e h\right )-\left (c e g-6 b f g+\frac {5 c f g^2}{h}-c d h+6 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^6} \, dx}{6 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac {\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}+\frac {\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{24 \left (c g^2-b g h+a h^2\right )^2}\\ &=\frac {\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac {\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}-\frac {\left (\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{(g+h x)^3} \, dx}{128 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{512 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}+\frac {\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac {\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{1024 \left (c g^2-b g h+a h^2\right )^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{512 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}+\frac {\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac {\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac {-b g+2 a h-(2 c g-b h) x}{\sqrt {a+b x+c x^2}}\right )}{512 \left (c g^2-b g h+a h^2\right )^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{512 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}+\frac {\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac {\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) \tanh ^{-1}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{1024 \left (c g^2-b g h+a h^2\right )^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.24, size = 766, normalized size = 1.17 \[ \frac {(a+x (b+c x))^{3/2} \left (-\frac {\frac {\left (a+b x+c x^2\right )^{5/2} \left (\frac {1}{2} c h \left (12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right )-c g \left (-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right )\right )}{5 (g+h x)^5 \left (a h^2-b g h+c g^2\right )}-\frac {\left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g)}{8 (g+h x)^4 \left (a h^2-b g h+c g^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a h-x (2 c g-b h)-b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right )}{2 \sqrt {a h^2-b g h+c g^2} \left (4 a h^2-4 b g h+4 c g^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\right )}{16 \left (a h^2-b g h+c g^2\right )}\right ) \left (b \left (-c g \left (-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right )-\frac {1}{2} c h \left (12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right )\right )-2 \left (-\frac {1}{2} c^2 g \left (12 h (a e h-a f g+c d g)-b h (7 d h+5 e g)+5 b f g^2\right )-a c h \left (-6 f h (b g-a h)+c h (e g-d h)+5 c f g^2\right )\right )\right )}{2 \left (a h^2-b g h+c g^2\right )}}{6 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (\frac {1}{2} h (-12 a f h+5 b f g+2 c d h)-\frac {1}{2} g (2 c (e h+5 f g)-7 b f h)\right )}{6 (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\right )}{c h \left (a+b x+c x^2\right )^{3/2}}-\frac {f \left (a+b x+c x^2\right ) (a+x (b+c x))^{3/2}}{c h (g+h x)^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.13, size = 100754, normalized size = 153.35 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________