Optimal. Leaf size=236 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8 d-2 \text {$\#$1}^4 a d+a^2 d+b^2 c\& ,\frac {-\text {$\#$1}^4 a d \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 a d \log (x)+a^2 d \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )+b^2 c \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )-a^2 d \log (x)-b^2 c \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{2 c d}-\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{c}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{c} \]
________________________________________________________________________________________
Rubi [B] time = 1.60, antiderivative size = 561, normalized size of antiderivative = 2.38, number of steps used = 17, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2056, 906, 63, 331, 298, 203, 206, 6725, 93, 205, 208} \begin {gather*} \frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}+\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {\sqrt [4]{a x^4-b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {-c}}}{\sqrt [8]{d} \sqrt [4]{a x-b}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{c x^{3/4} \sqrt [4]{a x-b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 203
Rule 205
Rule 206
Rule 208
Rule 298
Rule 331
Rule 906
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx &=\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{-b+a x}}{d+c x^2} \, dx}{x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {-a d-b c x}{\sqrt [4]{x} (-b+a x)^{3/4} \left (d+c x^2\right )} \, dx}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (a \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{c x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\sqrt [4]{-b x^3+a x^4} \int \left (\frac {-\frac {b c d}{\sqrt {-c}}-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\frac {b c d}{\sqrt {-c}}-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 a \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (4 a \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {-c} x\right )} \, dx}{2 c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (-b \sqrt {-c}+a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {-c} x\right )} \, dx}{2 c x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (2 \sqrt {a} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt {a} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (b \sqrt {-c}+a \sqrt {d}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \left (-b \sqrt {-c}+a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {d}-\left (-b \sqrt {-c}+a \sqrt {d}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}\\ &=-\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\sqrt {-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {-b \sqrt {-c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\sqrt {-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {-b \sqrt {-c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}-\sqrt {b \sqrt {-c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c \sqrt {b \sqrt {-c}+a \sqrt {d}} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (-b \sqrt {-c}-a \sqrt {d}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{d}+\sqrt {b \sqrt {-c}+a \sqrt {d}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c \sqrt {b \sqrt {-c}+a \sqrt {d}} x^{3/4} \sqrt [4]{-b+a x}}\\ &=-\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{b \sqrt {-c}+a \sqrt {d}} \sqrt [4]{x}}{\sqrt [8]{d} \sqrt [4]{-b+a x}}\right )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.24, size = 182, normalized size = 0.77 \begin {gather*} -\frac {2 \sqrt [4]{x^3 (a x-b)} \left (\left (b \sqrt {-c}-a \sqrt {d}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {\left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) x}{b-a x}\right )-\left (a \sqrt {d}+b \sqrt {-c}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {\left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) x}{b-a x}\right )+2 a \sqrt {d} \left (1-\frac {a x}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {a x}{b}\right )\right )}{3 c \sqrt {d} (b-a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.64, size = 235, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\frac {\text {RootSum}\left [b^2 c+a^2 d-2 a d \text {$\#$1}^4+d \text {$\#$1}^8\&,\frac {b^2 c \log (x)+a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )-a^2 d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right )-a d \log (x) \text {$\#$1}^4+a d \log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{2 c d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.78, size = 940, normalized size = 3.98 \begin {gather*} -2 \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (c^{7} d x \sqrt {-\frac {b^{2}}{c^{7} d}} - a c^{3} d x\right )} \sqrt {\frac {c^{2} x^{2} \sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {3}{4}} - {\left (c^{7} d \sqrt {-\frac {b^{2}}{c^{7} d}} - a c^{3} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c + a^{2} d\right )} x}\right ) + 2 \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (c^{7} d x \sqrt {-\frac {b^{2}}{c^{7} d}} + a c^{3} d x\right )} \sqrt {\frac {c^{2} x^{2} \sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {3}{4}} - {\left (c^{7} d \sqrt {-\frac {b^{2}}{c^{7} d}} + a c^{3} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c + a^{2} d\right )} x}\right ) - 4 \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {c^{3} x \sqrt {\frac {c^{2} x^{2} \sqrt {\frac {a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} c^{3} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}}}{a x}\right ) - \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}}}{c \,x^{2}+d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{c x^{2} + d}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{c\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (a x - b\right )}}{c x^{2} + d}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________