Optimal. Leaf size=339 \[ \frac {4 \sqrt {2} d^{7/4} ((a+b x) (c+d x))^{3/4} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right ),\frac {1}{2}\right )}{7 \sqrt [4]{b} (a+b x)^{3/4} (c+d x)^{3/4} (b c-a d)^{3/2} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}+\frac {8 d \sqrt [4]{c+d x}}{7 (a+b x)^{3/4} (b c-a d)^2}-\frac {4 \sqrt [4]{c+d x}}{7 (a+b x)^{7/4} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {51, 62, 623, 220} \[ \frac {4 \sqrt {2} d^{7/4} ((a+b x) (c+d x))^{3/4} \sqrt {(a d+b c+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{b} (a+b x)^{3/4} (c+d x)^{3/4} (b c-a d)^{3/2} (a d+b c+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}+\frac {8 d \sqrt [4]{c+d x}}{7 (a+b x)^{3/4} (b c-a d)^2}-\frac {4 \sqrt [4]{c+d x}}{7 (a+b x)^{7/4} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 62
Rule 220
Rule 623
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{11/4} (c+d x)^{3/4}} \, dx &=-\frac {4 \sqrt [4]{c+d x}}{7 (b c-a d) (a+b x)^{7/4}}-\frac {(6 d) \int \frac {1}{(a+b x)^{7/4} (c+d x)^{3/4}} \, dx}{7 (b c-a d)}\\ &=-\frac {4 \sqrt [4]{c+d x}}{7 (b c-a d) (a+b x)^{7/4}}+\frac {8 d \sqrt [4]{c+d x}}{7 (b c-a d)^2 (a+b x)^{3/4}}+\frac {\left (4 d^2\right ) \int \frac {1}{(a+b x)^{3/4} (c+d x)^{3/4}} \, dx}{7 (b c-a d)^2}\\ &=-\frac {4 \sqrt [4]{c+d x}}{7 (b c-a d) (a+b x)^{7/4}}+\frac {8 d \sqrt [4]{c+d x}}{7 (b c-a d)^2 (a+b x)^{3/4}}+\frac {\left (4 d^2 ((a+b x) (c+d x))^{3/4}\right ) \int \frac {1}{\left (a c+(b c+a d) x+b d x^2\right )^{3/4}} \, dx}{7 (b c-a d)^2 (a+b x)^{3/4} (c+d x)^{3/4}}\\ &=-\frac {4 \sqrt [4]{c+d x}}{7 (b c-a d) (a+b x)^{7/4}}+\frac {8 d \sqrt [4]{c+d x}}{7 (b c-a d)^2 (a+b x)^{3/4}}+\frac {\left (16 d^2 ((a+b x) (c+d x))^{3/4} \sqrt {(b c+a d+2 b d x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{7 (b c-a d)^2 (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x)}\\ &=-\frac {4 \sqrt [4]{c+d x}}{7 (b c-a d) (a+b x)^{7/4}}+\frac {8 d \sqrt [4]{c+d x}}{7 (b c-a d)^2 (a+b x)^{3/4}}+\frac {4 \sqrt {2} d^{7/4} ((a+b x) (c+d x))^{3/4} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{b} (b c-a d)^{3/2} (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 73, normalized size = 0.22 \[ -\frac {4 \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (-\frac {7}{4},\frac {3}{4};-\frac {3}{4};\frac {d (a+b x)}{a d-b c}\right )}{7 b (a+b x)^{7/4} (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b^{3} d x^{4} + a^{3} c + {\left (b^{3} c + 3 \, a b^{2} d\right )} x^{3} + 3 \, {\left (a b^{2} c + a^{2} b d\right )} x^{2} + {\left (3 \, a^{2} b c + a^{3} d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {11}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {11}{4}} \left (d x +c \right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {11}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,x\right )}^{11/4}\,{\left (c+d\,x\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x\right )^{\frac {11}{4}} \left (c + d x\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________