Optimal. Leaf size=245 \[ -\frac {a (d g+e f) \text {RootSum}\left [\text {$\#$1}^8 d-4 \text {$\#$1}^6 c d-2 \text {$\#$1}^4 b d+6 \text {$\#$1}^4 c^2 d+4 \text {$\#$1}^2 b c d-4 \text {$\#$1}^2 c^3 d+a^2 e+b^2 d-2 b c^2 d+c^4 d\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{\text {$\#$1}^5-2 \text {$\#$1}^3 c-\text {$\#$1} b+\text {$\#$1} c^2}\& \right ]}{2 d^2}+\frac {4 f \left (3 a x+3 b+8 c^2\right ) \sqrt {\sqrt {a x+b}+c}}{15 a d}-\frac {16 c f \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}}{15 a d} \]
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Rubi [B] time = 23.54, antiderivative size = 2139, normalized size of antiderivative = 8.73, number of steps used = 31, number of rules used = 10, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6740, 194, 6688, 12, 1988, 1094, 634, 618, 206, 628}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 194
Rule 206
Rule 618
Rule 628
Rule 634
Rule 1094
Rule 1988
Rule 6688
Rule 6740
Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (-a^2 g+f \left (b-x^2\right )^2\right )}{\sqrt {c+x} \left (e+\frac {d \left (b-x^2\right )^2}{a^2}\right )} \, dx,x,\sqrt {b+a x}\right )}{a^3}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {\left (c-x^2\right )^2 \left (-a^2 g+f \left (b-\left (c-x^2\right )^2\right )^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (\frac {a^2 b f}{d}-\frac {a^2 f \left (b-\left (c-x^2\right )^2\right )}{d}-\frac {a^2 b (e f+d g)-a^2 (e f+d g) \left (b-\left (c-x^2\right )^2\right )}{d \left (e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3}\\ &=\frac {4 b f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {4 \operatorname {Subst}\left (\int \frac {a^2 b (e f+d g)-a^2 (e f+d g) \left (b-\left (c-x^2\right )^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d}-\frac {(4 f) \operatorname {Subst}\left (\int \left (b-\left (c-x^2\right )^2\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=-\frac {4 \operatorname {Subst}\left (\int \frac {a^2 (e f+d g) \left (c-x^2\right )^2}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d}+\frac {(4 f) \operatorname {Subst}\left (\int \left (c-x^2\right )^2 \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=\frac {(4 f) \operatorname {Subst}\left (\int \left (c^2-2 c x^2+x^4\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}-\frac {(4 (e f+d g)) \operatorname {Subst}\left (\int \frac {\left (c-x^2\right )^2}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {(4 (e f+d g)) \operatorname {Subst}\left (\int \left (\frac {a b \sqrt {e}-\frac {a^2 e}{\sqrt {-d}}}{2 e \left (a \sqrt {e}-\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )\right )}+\frac {a b \sqrt {e}+\frac {a^2 e}{\sqrt {-d}}}{2 e \left (a \sqrt {e}+\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (2 \left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {e}+\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}-\frac {\left (2 \left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {e}-\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (2 \left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}+2 c \sqrt {-d} x^2-\sqrt {-d} x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}-\frac {\left (2 \left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}-2 c \sqrt {-d} x^2+\sqrt {-d} x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}-x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}+\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}-x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}+\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}+\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}+\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}+\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}+\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}-\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}+\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}-\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}+\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c-\frac {\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}{(-d)^{3/4}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c-\frac {\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}{(-d)^{3/4}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c+\frac {d \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{(-d)^{5/4}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c+\frac {d \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{(-d)^{5/4}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {(-d)^{3/8} \left (\frac {\sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}{(-d)^{3/8}}-\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {\sqrt [8]{-d} \left (\frac {\sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}-\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {(-d)^{3/8} \left (\frac {\sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}{(-d)^{3/8}}+\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {\sqrt [8]{-d} \left (\frac {\sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}-\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}+\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}-\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}+\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 27.37, size = 373, normalized size = 1.52 \begin {gather*} -\frac {\left (\sqrt {a x+b}+c\right )^{7/2} \left (\frac {c}{\sqrt {a x+b}+c}-1\right ) \left (\frac {15 a^2 (d g+e f) \text {RootSum}\left [\text {$\#$1}^8 a^2 e+\text {$\#$1}^8 b^2 d-2 \text {$\#$1}^8 b c^2 d+\text {$\#$1}^8 c^4 d+4 \text {$\#$1}^6 b c d-4 \text {$\#$1}^6 c^3 d-2 \text {$\#$1}^4 b d+6 \text {$\#$1}^4 c^2 d-4 \text {$\#$1}^2 c d+d\&,\frac {\text {$\#$1}^5 c^2 \log \left (\frac {1}{\sqrt {\sqrt {a x+b}+c}}-\text {$\#$1}\right )-2 \text {$\#$1}^3 c \log \left (\frac {1}{\sqrt {\sqrt {a x+b}+c}}-\text {$\#$1}\right )+\text {$\#$1} \log \left (\frac {1}{\sqrt {\sqrt {a x+b}+c}}-\text {$\#$1}\right )}{\text {$\#$1}^6 a^2 e+\text {$\#$1}^6 b^2 d-2 \text {$\#$1}^6 b c^2 d+\text {$\#$1}^6 c^4 d+3 \text {$\#$1}^4 b c d-3 \text {$\#$1}^4 c^3 d-\text {$\#$1}^2 b d+3 \text {$\#$1}^2 c^2 d-c d}\&\right ]}{\left (\sqrt {a x+b}+c\right )^{5/2}}+\frac {8 f \left (-4 c \sqrt {a x+b}+3 a x+3 b+8 c^2\right )}{\left (\sqrt {a x+b}+c\right )^2}\right )}{30 a d \sqrt {a x+b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 223, normalized size = 0.91 \begin {gather*} \frac {4 \sqrt {c+\sqrt {b+a x}} \left (8 c^2 f-4 c f \sqrt {b+a x}+3 f (b+a x)\right )}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b \text {$\#$1}+c^2 \text {$\#$1}-2 c \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 146.73, size = 2731, normalized size = 11.15
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 226, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {4 f \left (\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+c^{2} \sqrt {c +\sqrt {a x +b}}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +a^{2} e +b^{2} d \right )}{\sum }\frac {\left (\textit {\_R}^{4} \left (d g +e f \right )+2 c \left (-d g -e f \right ) \textit {\_R}^{2}+c^{2} d g +c^{2} e f \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\) | \(226\) |
default | \(\frac {\frac {4 f \left (\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+c^{2} \sqrt {c +\sqrt {a x +b}}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +a^{2} e +b^{2} d \right )}{\sum }\frac {\left (\textit {\_R}^{4} \left (d g +e f \right )+2 c \left (-d g -e f \right ) \textit {\_R}^{2}+c^{2} d g +c^{2} e f \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\) | \(226\) |
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 \begin {gather*} \int \frac {{\left (f x^{2} - g\right )} \sqrt {a x + b}}{{\left (d x^{2} + e\right )} \sqrt {c + \sqrt {a x + b}}}\,{d x} \end {gather*}
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 \begin {gather*} \int -\frac {\left (g-f\,x^2\right )\,\sqrt {b+a\,x}}{\sqrt {c+\sqrt {b+a\,x}}\,\left (d\,x^2+e\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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