Optimal. Leaf size=101 \[ -\frac {16 d^2 (c+d x)^{5/2}}{315 (a+b x)^{5/2} (b c-a d)^3}+\frac {8 d (c+d x)^{5/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} -\frac {16 d^2 (c+d x)^{5/2}}{315 (a+b x)^{5/2} (b c-a d)^3}+\frac {8 d (c+d x)^{5/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{9 (a+b x)^{9/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^{11/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}-\frac {(4 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx}{9 (b c-a d)}\\ &=-\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}+\frac {\left (8 d^2\right ) \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{63 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{5/2}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {8 d (c+d x)^{5/2}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {16 d^2 (c+d x)^{5/2}}{315 (b c-a d)^3 (a+b x)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 77, normalized size = 0.76 \begin {gather*} -\frac {2 (c+d x)^{5/2} \left (63 a^2 d^2+18 a b d (2 d x-5 c)+b^2 \left (35 c^2-20 c d x+8 d^2 x^2\right )\right )}{315 (a+b x)^{9/2} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 73, normalized size = 0.72 \begin {gather*} -\frac {2 (c+d x)^{9/2} \left (\frac {63 d^2 (a+b x)^2}{(c+d x)^2}-\frac {90 b d (a+b x)}{c+d x}+35 b^2\right )}{315 (a+b x)^{9/2} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 13.56, size = 426, normalized size = 4.22 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} d^{4} x^{4} + 35 \, b^{2} c^{4} - 90 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} - 4 \, {\left (b^{2} c d^{3} - 9 \, a b d^{4}\right )} x^{3} + 3 \, {\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (25 \, b^{2} c^{3} d - 72 \, a b c^{2} d^{2} + 63 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{315 \, {\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \, {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.94, size = 1394, normalized size = 13.80
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 105, normalized size = 1.04 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (8 b^{2} x^{2} d^{2}+36 a b \,d^{2} x -20 b^{2} c d x +63 a^{2} d^{2}-90 a b c d +35 b^{2} c^{2}\right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.11, size = 268, normalized size = 2.65 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {126\,a^2\,c^2\,d^2-180\,a\,b\,c^3\,d+70\,b^2\,c^4}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,\left (126\,a^2\,d^4-36\,a\,b\,c\,d^3+6\,b^2\,c^2\,d^2\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,d^4\,x^4}{315\,b^2\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d^3\,x^3\,\left (9\,a\,d-b\,c\right )}{315\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {4\,c\,d\,x\,\left (63\,a^2\,d^2-72\,a\,b\,c\,d+25\,b^2\,c^2\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^3}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________