Optimal. Leaf size=341 \[ -\frac{2 d \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{f \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}-\frac{2 (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{d \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 \sqrt{f} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 \sqrt{f} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.04115, antiderivative size = 341, normalized size of antiderivative = 1., 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 = {6725, 636, 1018, 1033, 724, 206} \[ -\frac{2 d \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{f \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}-\frac{2 (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{d \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 \sqrt{f} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 \sqrt{f} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6725
Rule 636
Rule 1018
Rule 1033
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx &=\int \left (-\frac{x}{f \left (a+b x+c x^2\right )^{3/2}}+\frac{d x}{f \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )}\right ) \, dx\\ &=-\frac{\int \frac{x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{f}+\frac{d \int \frac{x}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx}{f}\\ &=-\frac{2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt{a+b x+c x^2}}-\frac{2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}+\frac{(2 d) \int \frac{\frac{1}{2} b \left (b^2-4 a c\right ) d f-\frac{1}{2} \left (b^2-4 a c\right ) f (c d+a f) x}{\sqrt{a+b x+c x^2} \left (d-f x^2\right )} \, dx}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right )}\\ &=-\frac{2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt{a+b x+c x^2}}-\frac{2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}+\frac{d \int \frac{1}{\left (-\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d-b \sqrt{d} \sqrt{f}+a f\right )}+\frac{d \int \frac{1}{\left (\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d+b \sqrt{d} \sqrt{f}+a f\right )}\\ &=-\frac{2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt{a+b x+c x^2}}-\frac{2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}-\frac{d \operatorname{Subst}\left (\int \frac{1}{4 c d f-4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{b \sqrt{d} \sqrt{f}-2 a f-\left (-2 c \sqrt{d} \sqrt{f}+b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{c d-b \sqrt{d} \sqrt{f}+a f}-\frac{d \operatorname{Subst}\left (\int \frac{1}{4 c d f+4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{-b \sqrt{d} \sqrt{f}-2 a f-\left (2 c \sqrt{d} \sqrt{f}+b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{c d+b \sqrt{d} \sqrt{f}+a f}\\ &=-\frac{2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt{a+b x+c x^2}}-\frac{2 d \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) x\right )}{\left (b^2-4 a c\right ) f \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}-\frac{d \tanh ^{-1}\left (\frac{b \sqrt{d}-2 a \sqrt{f}+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x}{2 \sqrt{c d-b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{f} \left (c d-b \sqrt{d} \sqrt{f}+a f\right )^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{b \sqrt{d}+2 a \sqrt{f}+\left (2 c \sqrt{d}+b \sqrt{f}\right ) x}{2 \sqrt{c d+b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{f} \left (c d+b \sqrt{d} \sqrt{f}+a f\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.37873, size = 414, normalized size = 1.21 \[ \frac{1}{2} \left (\frac{4 a^2 (b f x+2 c d)+8 a^3 f-4 a b d (b-3 c x)-4 b^3 d x}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (f \left (b^2 d-a^2 f\right )-2 a c d f-c^2 d^2\right )}-\frac{d \log \left (\sqrt{d} \sqrt{f}-f x\right )}{\sqrt{f} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}}-\frac{d \log \left (\sqrt{d} \sqrt{f}+f x\right )}{\sqrt{f} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{d \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )}{\sqrt{f} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{d \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{\sqrt{f} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.287, size = 1480, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]