Optimal. Leaf size=1125 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.31048, antiderivative size = 1125, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {957, 744, 834, 843, 718, 424, 419, 21, 934, 169, 538, 537} \[ -\frac{\sqrt{2} \sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right ) e^2}{\sqrt{c} (e f-d g)^3 \sqrt{c x^2+b x+a}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} g \sqrt{f+g 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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right ) e}{(e f-d g)^2 \left (c f^2-b g f+a g^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}+\frac{2 g^2 \sqrt{c x^2+b x+a} e}{(e f-d g)^2 \left (c f^2-b g f+a g^2\right ) \sqrt{f+g x}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} g (2 c f-b g) \sqrt{f+g 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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} g \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) \sqrt{f+g x} \sqrt{c x^2+b x+a}}+\frac{4 g^2 (2 c f-b g) \sqrt{c x^2+b x+a}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 g^2 \sqrt{c x^2+b x+a}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 957
Rule 744
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rule 21
Rule 934
Rule 169
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) (f+g x)^{5/2} \sqrt{a+b x+c x^2}} \, dx &=\int \left (-\frac{g}{(e f-d g) (f+g x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{e g}{(e f-d g)^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2}}+\frac{e^2}{(e f-d g)^2 (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}}\right ) \, dx\\ &=\frac{e^2 \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{(e f-d g)^2}-\frac{(e g) \int \frac{1}{(f+g x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{(e f-d g)^2}-\frac{g \int \frac{1}{(f+g x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{e f-d g}\\ &=\frac{2 g^2 \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac{2 e g^2 \sqrt{a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}+\frac{(2 e g) \int \frac{-\frac{c f}{2}-\frac{c g x}{2}}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )}+\frac{(2 g) \int \frac{\frac{1}{2} (-3 c f+2 b g)+\frac{c g x}{2}}{(f+g x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )}+\frac{\left (e^2 \sqrt{b-\sqrt{b^2-4 a c}+2 c x} \sqrt{b+\sqrt{b^2-4 a c}+2 c x}\right ) \int \frac{1}{\sqrt{b-\sqrt{b^2-4 a c}+2 c x} \sqrt{b+\sqrt{b^2-4 a c}+2 c x} (d+e x) \sqrt{f+g x}} \, dx}{(e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac{4 g^2 (2 c f-b g) \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}-\frac{(4 g) \int \frac{\frac{1}{4} c \left (3 c f^2-g (b f+a g)\right )+\frac{1}{2} c g (2 c f-b g) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2}-\frac{(c e g) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )}-\frac{\left (2 e^2 \sqrt{b-\sqrt{b^2-4 a c}+2 c x} \sqrt{b+\sqrt{b^2-4 a c}+2 c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (e f-d g-e x^2\right ) \sqrt{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}+\frac{2 c x^2}{g}} \sqrt{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}+\frac{2 c x^2}{g}}} \, dx,x,\sqrt{f+g x}\right )}{(e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac{4 g^2 (2 c f-b g) \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}-\frac{(2 c g (2 c f-b g)) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2}+\frac{(c g) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} e g \sqrt{f+g 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} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}}{\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 )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{a+b x+c x^2}}-\frac{\left (2 e^2 \sqrt{b+\sqrt{b^2-4 a c}+2 c x} \sqrt{1+\frac{2 c (f+g x)}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (e f-d g-e x^2\right ) \sqrt{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}+\frac{2 c x^2}{g}} \sqrt{1+\frac{2 c x^2}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}} \, dx,x,\sqrt{f+g x}\right )}{(e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac{4 g^2 (2 c f-b g) \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} e g \sqrt{f+g 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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}-\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} g (2 c f-b g) \sqrt{f+g 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} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}}{\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 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{a+b x+c x^2}}+\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} g \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \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} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}} \, 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 (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\left (2 e^2 \sqrt{1+\frac{2 c (f+g x)}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}} \sqrt{1+\frac{2 c (f+g x)}{\left (b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (e f-d g-e x^2\right ) \sqrt{1+\frac{2 c x^2}{\left (b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}} \sqrt{1+\frac{2 c x^2}{\left (b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}\right ) g}}} \, dx,x,\sqrt{f+g x}\right )}{(e f-d g)^2 \sqrt{a+b x+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^{3/2}}+\frac{4 g^2 (2 c f-b g) \sqrt{a+b x+c x^2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+b x+c x^2}}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} g (2 c f-b g) \sqrt{f+g 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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} e g \sqrt{f+g 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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} g \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} e^2 \sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{c} (e f-d g)^3 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 15.8711, size = 14762, normalized size = 13.12 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.46, size = 27597, normalized size = 24.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]