Optimal. Leaf size=222 \[ -\frac {6 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{\sqrt {a+b x} (b c-a d)^{5/4}}-\frac {6 d \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)}+\frac {6 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt {a+b x} (b c-a d)^{5/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {51, 63, 307, 224, 221, 1200, 1199, 424} \[ -\frac {6 d \sqrt {a+b x}}{\sqrt [4]{c+d x} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)}-\frac {6 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt {a+b x} (b c-a d)^{5/4}}+\frac {6 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt {a+b x} (b c-a d)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 221
Rule 224
Rule 307
Rule 424
Rule 1199
Rule 1200
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{3/2} (c+d x)^{5/4}} \, dx &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {(3 d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}} \, dx}{2 (b c-a d)}\\ &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {6 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {(3 b d) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{2 (b c-a d)^2}\\ &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {6 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{(b c-a d)^2}\\ &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {6 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt [4]{c+d x}}-\frac {\left (6 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{(b c-a d)^{3/2}}+\frac {\left (6 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{(b c-a d)^{3/2}}\\ &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {6 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt [4]{c+d x}}-\frac {\left (6 \sqrt {b} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{(b c-a d)^{3/2} \sqrt {a+b x}}+\frac {\left (6 \sqrt {b} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{(b c-a d)^{3/2} \sqrt {a+b x}}\\ &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {6 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt [4]{c+d x}}-\frac {6 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{(b c-a d)^{5/4} \sqrt {a+b x}}+\frac {\left (6 \sqrt {b} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{(b c-a d)^{3/2} \sqrt {a+b x}}\\ &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}-\frac {6 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt [4]{c+d x}}+\frac {6 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{(b c-a d)^{5/4} \sqrt {a+b x}}-\frac {6 \sqrt [4]{b} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{(b c-a d)^{5/4} \sqrt {a+b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 71, normalized size = 0.32 \[ -\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (-\frac {1}{2},\frac {5}{4};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt {a+b x} (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{4}}}{b^{2} d^{2} x^{4} + a^{2} c^{2} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} x^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b c^{2} + a^{2} c d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________