Optimal. Leaf size=66 \[ \frac {4 b \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^2}+\frac {2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac {4 b \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^2}+\frac {2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx &=\frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {(2 b) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)}\\ &=\frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 (b c-a d)^2 \sqrt {c+d x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 46, normalized size = 0.70 \[ \frac {2 \sqrt {a+b x} (-a d+3 b c+2 b d x)}{3 (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.02, size = 118, normalized size = 1.79 \[ \frac {2 \, {\left (2 \, b d x + 3 \, b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.12, size = 126, normalized size = 1.91 \[ \frac {2 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{4} d^{2}}{b^{2} c^{2} d {\left | b \right |} - 2 \, a b c d^{2} {\left | b \right |} + a^{2} d^{3} {\left | b \right |}} + \frac {3 \, {\left (b^{5} c d - a b^{4} d^{2}\right )}}{b^{2} c^{2} d {\left | b \right |} - 2 \, a b c d^{2} {\left | b \right |} + a^{2} d^{3} {\left | b \right |}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 53, normalized size = 0.80 \[ -\frac {2 \sqrt {b x +a}\, \left (-2 b d x +a d -3 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.44, size = 127, normalized size = 1.92 \[ \frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (6\,c\,b^2+2\,a\,d\,b\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,a^2\,d-6\,a\,b\,c}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,b^2\,x^2}{3\,d\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________