Optimal. Leaf size=144 \[ -\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {16 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{7 \sqrt [4]{b} d^3 \sqrt {a+b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 144, 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 = {52, 65, 230,
227} \begin {gather*} \frac {16 (b c-a d)^{9/4} \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 )}{7 \sqrt [4]{b} d^3 \sqrt {a+b x}}-\frac {8 \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 227
Rule 230
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx &=\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}-\frac {(6 (b c-a d)) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}} \, dx}{7 d}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {\left (4 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{7 d^2}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {\left (16 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{7 d^3}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {\left (16 (b c-a d)^2 \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {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 )}{7 d^3 \sqrt {a+b x}}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {16 (b c-a d)^{9/4} \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 )}{7 \sqrt [4]{b} d^3 \sqrt {a+b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 73, normalized size = 0.51 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{-b c+a d}\right )}{5 b (c+d x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {3}{2}}}{\left (d x +c \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\left (a + b x\right )^{\frac {3}{2}}}{\left (c + d x\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________