Optimal. Leaf size=730 \[ -\frac{16 c^3 \left (8 a d^2+c^3\right ) \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{35 d^2 \sqrt{4 a d^2+c^3} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}+\frac{2 c \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac{c}{d}+x\right )^2\right )}{35 d^2}+\frac{1}{7} \left (\frac{c}{d}+x\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}+\frac{8 c^{7/4} \left (4 a d^2+c^3\right )^{3/4} \left (\sqrt{4 a d^2+c^3} \left (5 a d^2+c^3\right )-c^{3/2} \left (8 a d^2+c^3\right )\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{16 c^{13/4} \left (4 a d^2+c^3\right )^{3/4} \left (8 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.903018, antiderivative size = 730, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {1106, 1091, 1176, 1197, 1103, 1195} \[ -\frac{16 c^3 \left (8 a d^2+c^3\right ) \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{35 d^2 \sqrt{4 a d^2+c^3} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )}+\frac{2 c \left (\frac{c}{d}+x\right ) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (20 a d^2+7 c^3-3 c d^2 \left (\frac{c}{d}+x\right )^2\right )}{35 d^2}+\frac{1}{7} \left (\frac{c}{d}+x\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}+\frac{8 c^{7/4} \left (4 a d^2+c^3\right )^{3/4} \left (\sqrt{4 a d^2+c^3} \left (5 a d^2+c^3\right )-c^{3/2} \left (8 a d^2+c^3\right )\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{16 c^{13/4} \left (4 a d^2+c^3\right )^{3/4} \left (8 a d^2+c^3\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right )^2}} \left (\frac{d^2 \left (\frac{c}{d}+x\right )^2}{\sqrt{4 a d^2+c^3}}+\sqrt{c}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}+1\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1106
Rule 1091
Rule 1176
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \left (c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4\right )^{3/2} \, dx,x,\frac{c}{d}+x\right )\\ &=\frac{1}{7} \left (\frac{c}{d}+x\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}+\frac{3}{7} \operatorname{Subst}\left (\int \left (2 c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2\right ) \sqrt{c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4} \, dx,x,\frac{c}{d}+x\right )\\ &=\frac{1}{7} \left (\frac{c}{d}+x\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}+\frac{2 c (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (7 c^3+20 a d^2-3 c (c+d x)^2\right )}{35 d^3}+\frac{\operatorname{Subst}\left (\int \frac{\frac{16 c^2 \left (c^3+4 a d^2\right ) \left (c^3+5 a d^2\right )}{d^2}-16 c^3 \left (c^3+8 a d^2\right ) x^2}{\sqrt{c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4}} \, dx,x,\frac{c}{d}+x\right )}{35 d^2}\\ &=\frac{1}{7} \left (\frac{c}{d}+x\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}+\frac{2 c (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (7 c^3+20 a d^2-3 c (c+d x)^2\right )}{35 d^3}+\frac{\left (16 c^{7/2} \sqrt{c^3+4 a d^2} \left (c^3+8 a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1-\frac{d^2 x^2}{\sqrt{c} \sqrt{c^3+4 a d^2}}}{\sqrt{c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4}} \, dx,x,\frac{c}{d}+x\right )}{35 d^4}+\frac{\left (16 c^2 \sqrt{c^3+4 a d^2} \left (\sqrt{c^3+4 a d^2} \left (c^3+5 a d^2\right )-c^{3/2} \left (c^3+8 a d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4}} \, dx,x,\frac{c}{d}+x\right )}{35 d^4}\\ &=\frac{1}{7} \left (\frac{c}{d}+x\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^{3/2}+\frac{2 c (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \left (7 c^3+20 a d^2-3 c (c+d x)^2\right )}{35 d^3}-\frac{16 c^3 \left (c^3+8 a d^2\right ) (c+d x) \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{35 d^3 \sqrt{c^3+4 a d^2} \left (\sqrt{c}+\frac{(c+d x)^2}{\sqrt{c^3+4 a d^2}}\right )}+\frac{16 c^{13/4} \left (c^3+4 a d^2\right )^{3/4} \left (c^3+8 a d^2\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt{c}+\frac{(c+d x)^2}{\sqrt{c^3+4 a d^2}}\right )^2}} \left (\sqrt{c}+\frac{(c+d x)^2}{\sqrt{c^3+4 a d^2}}\right ) E\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (1+\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac{8 c^{7/4} \left (c^3+4 a d^2\right )^{3/4} \left (\sqrt{c^3+4 a d^2} \left (c^3+5 a d^2\right )-c^{3/2} \left (c^3+8 a d^2\right )\right ) \sqrt{\frac{d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt{c}+\frac{(c+d x)^2}{\sqrt{c^3+4 a d^2}}\right )^2}} \left (\sqrt{c}+\frac{(c+d x)^2}{\sqrt{c^3+4 a d^2}}\right ) F\left (2 \tan ^{-1}\left (\frac{c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac{1}{2} \left (1+\frac{c^{3/2}}{\sqrt{c^3+4 a d^2}}\right )\right )}{35 d^5 \sqrt{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}\\ \end{align*}
Mathematica [C] time = 6.19849, size = 10468, normalized size = 14.34 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.181, size = 5229, normalized size = 7.2 \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{\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (4 a c + 4 c^{2} x^{2} + 4 c d x^{3} + d^{2} x^{4}\right )^{\frac{3}{2}}\, dx \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*} \int{\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]