Optimal. Leaf size=893 \[ \frac {80 \sqrt {a+b x} d^2}{9 (b c-a d)^3 \sqrt [6]{c+d x}}+\frac {80 \left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt {a+b x} \sqrt [6]{c+d x} d^2}{9 (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {80 \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) d}{3\ 3^{3/4} (b c-a d)^{8/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {40 \left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) d}{9 \sqrt [4]{3} (b c-a d)^{8/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {20 d}{9 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.87, antiderivative size = 893, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {51, 63, 308, 225, 1881} \[ \frac {80 \sqrt {a+b x} d^2}{9 (b c-a d)^3 \sqrt [6]{c+d x}}+\frac {80 \left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt {a+b x} \sqrt [6]{c+d x} d^2}{9 (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {80 \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) d}{3\ 3^{3/4} (b c-a d)^{8/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {40 \left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) d}{9 \sqrt [4]{3} (b c-a d)^{8/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {20 d}{9 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 225
Rule 308
Rule 1881
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/6}} \, dx &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}-\frac {(10 d) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{7/6}} \, dx}{9 (b c-a d)}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}+\frac {20 d}{9 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}+\frac {\left (40 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/6}} \, dx}{27 (b c-a d)^2}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}+\frac {20 d}{9 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}+\frac {80 d^2 \sqrt {a+b x}}{9 (b c-a d)^3 \sqrt [6]{c+d x}}-\frac {\left (80 b d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}} \, dx}{27 (b c-a d)^3}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}+\frac {20 d}{9 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}+\frac {80 d^2 \sqrt {a+b x}}{9 (b c-a d)^3 \sqrt [6]{c+d x}}-\frac {(160 b d) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{9 (b c-a d)^3}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}+\frac {20 d}{9 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}+\frac {80 d^2 \sqrt {a+b x}}{9 (b c-a d)^3 \sqrt [6]{c+d x}}+\frac {\left (80 \sqrt [3]{b} d\right ) \operatorname {Subst}\left (\int \frac {\left (-1+\sqrt {3}\right ) (b c-a d)^{2/3}-2 b^{2/3} x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{9 (b c-a d)^3}+\frac {\left (80 \left (1-\sqrt {3}\right ) \sqrt [3]{b} d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{9 (b c-a d)^{7/3}}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}+\frac {20 d}{9 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}+\frac {80 d^2 \sqrt {a+b x}}{9 (b c-a d)^3 \sqrt [6]{c+d x}}+\frac {80 \left (1+\sqrt {3}\right ) \sqrt [3]{b} d^2 \sqrt {a+b x} \sqrt [6]{c+d x}}{9 (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {80 \sqrt [3]{b} d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3\ 3^{3/4} (b c-a d)^{8/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {40 \left (1-\sqrt {3}\right ) \sqrt [3]{b} d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9 \sqrt [4]{3} (b c-a d)^{8/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 73, normalized size = 0.08 \[ -\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \, _2F_1\left (-\frac {3}{2},\frac {7}{6};-\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/2} (c+d x)^{7/6}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{6}}}{b^{3} d^{2} x^{5} + a^{3} c^{2} + {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{4} + {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} + {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {7}{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{7/6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {7}{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________