Optimal. Leaf size=229 \[ -\frac {8 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}+\frac {16 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt {a+b x}}-\frac {16 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt {a+b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 229, 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 = {52, 65, 313,
230, 227, 1214, 1213, 435} \begin {gather*} -\frac {16 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt {a+b x}}+\frac {16 (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{3/4} d^3 \sqrt {a+b x}}-\frac {8 \sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 227
Rule 230
Rule 313
Rule 435
Rule 1213
Rule 1214
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{\sqrt [4]{c+d x}} \, dx &=\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {(2 (b c-a d)) \int \frac {\sqrt {a+b x}}{\sqrt [4]{c+d x}} \, dx}{3 d}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}+\frac {\left (4 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{15 d^2}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}+\frac {\left (16 (b c-a d)^2\right ) \text {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 )}{15 d^3}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {\left (16 (b c-a d)^{5/2}\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 )}{15 \sqrt {b} d^3}+\frac {\left (16 (b c-a d)^{5/2}\right ) \text {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 )}{15 \sqrt {b} d^3}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {\left (16 (b c-a d)^{5/2} \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 )}{15 \sqrt {b} d^3 \sqrt {a+b x}}+\frac {\left (16 (b c-a d)^{5/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {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 )}{15 \sqrt {b} d^3 \sqrt {a+b x}}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {16 (b c-a d)^{11/4} \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 )}{15 b^{3/4} d^3 \sqrt {a+b x}}+\frac {\left (16 (b c-a d)^{5/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {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 )}{15 \sqrt {b} d^3 \sqrt {a+b x}}\\ &=-\frac {8 (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{15 d^2}+\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}+\frac {16 (b c-a d)^{11/4} \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 )}{15 b^{3/4} d^3 \sqrt {a+b x}}-\frac {16 (b c-a d)^{11/4} \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 )}{15 b^{3/4} 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 = 0.32 \begin {gather*} \frac {2 (a+b x)^{5/2} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{4},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{-b c+a d}\right )}{5 b \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {3}{2}}}{\left (d x +c \right )^{\frac {1}{4}}}\, dx\]
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 = 0.00, 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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {3}{2}}}{\sqrt [4]{c + d x}}\, dx \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*} \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________