Optimal. Leaf size=126 \[ -\frac {2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac {2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac {2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e) (d+e x)^{7/2}}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{9/2}}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^{11/2}}{e^3}+\frac {B c (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac {2 d (B d-A e) (c d-b e) (d+e x)^{9/2}}{9 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{13/2}}{13 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 113, normalized size = 0.90 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (5 A e \left (13 b e (9 e x-2 d)+c \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+B \left (5 b e \left (8 d^2-36 d e x+99 e^2 x^2\right )+c \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.09, size = 141, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (585 A b e^2 (d+e x)-715 A b d e^2+715 A c d^2 e-1170 A c d e (d+e x)+495 A c e (d+e x)^2+715 b B d^2 e-1170 b B d e (d+e x)+495 b B e (d+e x)^2-715 B c d^3+1755 B c d^2 (d+e x)-1485 B c d (d+e x)^2+429 B c (d+e x)^3\right )}{6435 e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 271, normalized size = 2.15 \begin {gather*} \frac {2 \, {\left (429 \, B c e^{7} x^{7} - 16 \, B c d^{7} - 130 \, A b d^{5} e^{2} + 40 \, {\left (B b + A c\right )} d^{6} e + 33 \, {\left (46 \, B c d e^{6} + 15 \, {\left (B b + A c\right )} e^{7}\right )} x^{6} + 9 \, {\left (206 \, B c d^{2} e^{5} + 65 \, A b e^{7} + 200 \, {\left (B b + A c\right )} d e^{6}\right )} x^{5} + 10 \, {\left (80 \, B c d^{3} e^{4} + 221 \, A b d e^{6} + 229 \, {\left (B b + A c\right )} d^{2} e^{5}\right )} x^{4} + 5 \, {\left (B c d^{4} e^{3} + 598 \, A b d^{2} e^{5} + 212 \, {\left (B b + A c\right )} d^{3} e^{4}\right )} x^{3} - 3 \, {\left (2 \, B c d^{5} e^{2} - 520 \, A b d^{3} e^{4} - 5 \, {\left (B b + A c\right )} d^{4} e^{3}\right )} x^{2} + {\left (8 \, B c d^{6} e + 65 \, A b d^{4} e^{3} - 20 \, {\left (B b + A c\right )} d^{5} e^{2}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.24, size = 1383, normalized size = 10.98
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 121, normalized size = 0.96 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (-429 B c \,x^{3} e^{3}-495 A c \,e^{3} x^{2}-495 B b \,e^{3} x^{2}+198 B c d \,e^{2} x^{2}-585 A b \,e^{3} x +180 A c d \,e^{2} x +180 B b d \,e^{2} x -72 B c \,d^{2} e x +130 A b d \,e^{2}-40 A c \,d^{2} e -40 B b \,d^{2} e +16 B c \,d^{3}\right )}{6435 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.57, size = 112, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} B c - 495 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 585 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 111, normalized size = 0.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{11\,e^4}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{13\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.27, size = 683, normalized size = 5.42 \begin {gather*} \begin {cases} - \frac {4 A b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 A b d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 A b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 A b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 A b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 A b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 A c d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 A c d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 A c d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 A c d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 A c d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 A c d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 A c e^{3} x^{6} \sqrt {d + e x}}{13} + \frac {16 B b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 B b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 B b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 B b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 B b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 B b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 B b e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 B c d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 B c d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 B c d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 B c d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 B c d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 B c d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B c d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 B c e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {A b x^{2}}{2} + \frac {A c x^{3}}{3} + \frac {B b x^{3}}{3} + \frac {B c x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________