Optimal. Leaf size=341 \[ -\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {e \sqrt {b x+c x^2} \left (2 b^3 c e^2 (3 A e+7 B d)+4 b^2 c^2 d e (A e+2 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{3 b^4 c^3}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.46, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {818, 640, 620, 206} \begin {gather*} -\frac {e \sqrt {b x+c x^2} \left (4 b^2 c^2 d e (A e+2 B d)+2 b^3 c e^2 (3 A e+7 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{3 b^4 c^3}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (4 b^2 c^2 d e (A e+B d)+2 b^3 c e^2 (A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 620
Rule 640
Rule 818
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {(d+e x)^2 \left (\frac {1}{2} d \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )+\frac {1}{2} e \left (4 A c^2 d+5 b^2 B e-2 b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b d e \left (16 A c^3 d^2-5 b^3 B e^2+2 b^2 c e (2 B d+A e)-8 b c^2 d (B d+3 A e)\right )-\frac {1}{4} e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) x}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 c^3}+\frac {\left (e^3 (8 B c d-5 b B e+2 A c e)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c^3}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 c^3}+\frac {\left (e^3 (8 B c d-5 b B e+2 A c e)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^3}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 c^3}+\frac {e^3 (8 B c d-5 b B e+2 A c e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.37, size = 237, normalized size = 0.70 \begin {gather*} \frac {x^{5/2} \left (\frac {e^3 (b+c x)^{5/2} \log \left (\sqrt {c} \sqrt {b+c x}+c \sqrt {x}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}}-\frac {(b+c x) \left (2 x^2 (b+c x) (c d-b e)^3 \left (b c (5 B d-4 A e)-8 A c^2 d+7 b^2 B e\right )+2 c^3 d^3 x (b+c x)^2 (3 b (4 A e+B d)-8 A c d)+2 b x^2 (b B-A c) (c d-b e)^4+2 A b c^3 d^4 (b+c x)^2-3 b^4 B e^4 x^2 (b+c x)^2\right )}{3 b^4 c^3 x^{3/2}}\right )}{(x (b+c x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.03, size = 435, normalized size = 1.28 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-6 A b^5 c e^4 x^2-8 A b^4 c^2 e^4 x^3-2 A b^3 c^3 d^4-24 A b^3 c^3 d^3 e x+36 A b^3 c^3 d^2 e^2 x^2+8 A b^3 c^3 d e^3 x^3+12 A b^2 c^4 d^4 x-96 A b^2 c^4 d^3 e x^2+24 A b^2 c^4 d^2 e^2 x^3+48 A b c^5 d^4 x^2-64 A b c^5 d^3 e x^3+32 A c^6 d^4 x^3+15 b^6 B e^4 x^2-24 b^5 B c d e^3 x^2+20 b^5 B c e^4 x^3-32 b^4 B c^2 d e^3 x^3+3 b^4 B c^2 e^4 x^4-6 b^3 B c^3 d^4 x+24 b^3 B c^3 d^3 e x^2+12 b^3 B c^3 d^2 e^2 x^3-24 b^2 B c^4 d^4 x^2+16 b^2 B c^4 d^3 e x^3-16 b B c^5 d^4 x^3\right )}{3 b^4 c^3 x^2 (b+c x)^2}+\frac {\log \left (-2 c^{7/2} \sqrt {b x+c x^2}+b c^3+2 c^4 x\right ) \left (-2 A c e^4+5 b B e^4-8 B c d e^3\right )}{2 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 974, normalized size = 2.86 \begin {gather*} \left [\frac {3 \, {\left ({\left (8 \, B b^{4} c^{3} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{4} + 2 \, {\left (8 \, B b^{5} c^{2} d e^{3} - {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{3} + {\left (8 \, B b^{6} c d e^{3} - {\left (5 \, B b^{7} - 2 \, A b^{6} c\right )} e^{4}\right )} x^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (3 \, B b^{4} c^{3} e^{4} x^{4} - 2 \, A b^{3} c^{4} d^{4} - 4 \, {\left (4 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} - 4 \, {\left (B b^{2} c^{5} - 4 \, A b c^{6}\right )} d^{3} e - 3 \, {\left (B b^{3} c^{4} + 2 \, A b^{2} c^{5}\right )} d^{2} e^{2} + 2 \, {\left (4 \, B b^{4} c^{3} - A b^{3} c^{4}\right )} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (12 \, A b^{3} c^{4} d^{2} e^{2} - 8 \, B b^{5} c^{2} d e^{3} - 8 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 8 \, {\left (B b^{3} c^{4} - 4 \, A b^{2} c^{5}\right )} d^{3} e + {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{2} - 6 \, {\left (4 \, A b^{3} c^{4} d^{3} e + {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{6 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}}, -\frac {3 \, {\left ({\left (8 \, B b^{4} c^{3} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{4} + 2 \, {\left (8 \, B b^{5} c^{2} d e^{3} - {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{3} + {\left (8 \, B b^{6} c d e^{3} - {\left (5 \, B b^{7} - 2 \, A b^{6} c\right )} e^{4}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (3 \, B b^{4} c^{3} e^{4} x^{4} - 2 \, A b^{3} c^{4} d^{4} - 4 \, {\left (4 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} - 4 \, {\left (B b^{2} c^{5} - 4 \, A b c^{6}\right )} d^{3} e - 3 \, {\left (B b^{3} c^{4} + 2 \, A b^{2} c^{5}\right )} d^{2} e^{2} + 2 \, {\left (4 \, B b^{4} c^{3} - A b^{3} c^{4}\right )} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (12 \, A b^{3} c^{4} d^{2} e^{2} - 8 \, B b^{5} c^{2} d e^{3} - 8 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 8 \, {\left (B b^{3} c^{4} - 4 \, A b^{2} c^{5}\right )} d^{3} e + {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{2} - 6 \, {\left (4 \, A b^{3} c^{4} d^{3} e + {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 370, normalized size = 1.09 \begin {gather*} -\frac {\frac {2 \, A d^{4}}{b} - {\left ({\left ({\left (\frac {3 \, B x e^{4}}{c} - \frac {4 \, {\left (4 \, B b c^{5} d^{4} - 8 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 16 \, A b c^{5} d^{3} e - 3 \, B b^{3} c^{3} d^{2} e^{2} - 6 \, A b^{2} c^{4} d^{2} e^{2} + 8 \, B b^{4} c^{2} d e^{3} - 2 \, A b^{3} c^{3} d e^{3} - 5 \, B b^{5} c e^{4} + 2 \, A b^{4} c^{2} e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac {3 \, {\left (8 \, B b^{2} c^{4} d^{4} - 16 \, A b c^{5} d^{4} - 8 \, B b^{3} c^{3} d^{3} e + 32 \, A b^{2} c^{4} d^{3} e - 12 \, A b^{3} c^{3} d^{2} e^{2} + 8 \, B b^{5} c d e^{3} - 5 \, B b^{6} e^{4} + 2 \, A b^{5} c e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac {6 \, {\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )}}{b^{4} c^{3}}\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} - \frac {{\left (8 \, B c d e^{3} - 5 \, B b e^{4} + 2 \, A c e^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 1026, normalized size = 3.01 \begin {gather*} \frac {B \,e^{4} x^{4}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {A \,e^{4} x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {5 B b \,e^{4} x^{3}}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {4 B d \,e^{3} x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {A b \,e^{4} x^{2}}{2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {4 A d \,e^{3} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {5 B \,b^{2} e^{4} x^{2}}{4 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}+\frac {2 B b d \,e^{3} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {6 B \,d^{2} e^{2} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {A \,b^{2} e^{4} x}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}-\frac {4 A b d \,e^{3} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {8 A \,d^{3} e x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}-\frac {4 A c \,d^{4} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2}}-\frac {4 A \,d^{2} e^{2} x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {5 B \,b^{3} e^{4} x}{12 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{4}}+\frac {2 B \,b^{2} d \,e^{3} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}-\frac {2 B b \,d^{2} e^{2} x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {2 B \,d^{4} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}-\frac {8 B \,d^{3} e x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {8 A d \,e^{3} x}{3 \sqrt {c \,x^{2}+b x}\, b c}-\frac {2 A \,d^{4}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}+\frac {8 A \,d^{2} e^{2} x}{\sqrt {c \,x^{2}+b x}\, b^{2}}-\frac {64 A c \,d^{3} e x}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}+\frac {32 A \,c^{2} d^{4} x}{3 \sqrt {c \,x^{2}+b x}\, b^{4}}-\frac {7 A \,e^{4} x}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {35 B b \,e^{4} x}{6 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {4 B \,d^{2} e^{2} x}{\sqrt {c \,x^{2}+b x}\, b c}+\frac {16 B \,d^{3} e x}{3 \sqrt {c \,x^{2}+b x}\, b^{2}}-\frac {16 B c \,d^{4} x}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}-\frac {28 B d \,e^{3} x}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {A \,e^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}-\frac {5 B b \,e^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {7}{2}}}+\frac {4 B d \,e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}-\frac {A b \,e^{4}}{6 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {4 A \,d^{2} e^{2}}{\sqrt {c \,x^{2}+b x}\, b c}-\frac {32 A \,d^{3} e}{3 \sqrt {c \,x^{2}+b x}\, b^{2}}+\frac {16 A c \,d^{4}}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}+\frac {4 A d \,e^{3}}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {5 B \,b^{2} e^{4}}{12 \sqrt {c \,x^{2}+b x}\, c^{4}}-\frac {2 B b d \,e^{3}}{3 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {8 B \,d^{3} e}{3 \sqrt {c \,x^{2}+b x}\, b c}-\frac {8 B \,d^{4}}{3 \sqrt {c \,x^{2}+b x}\, b^{2}}+\frac {2 B \,d^{2} e^{2}}{\sqrt {c \,x^{2}+b x}\, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.65, size = 795, normalized size = 2.33 \begin {gather*} \frac {5 \, B b e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )}}{6 \, c} + \frac {B e^{4} x^{4}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {4 \, A c d^{4} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d^{4} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {10 \, B b e^{4} x}{3 \, \sqrt {c x^{2} + b x} c^{3}} - \frac {5 \, B b e^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {7}{2}}} - \frac {1}{3} \, {\left (4 \, B d e^{3} + A e^{4}\right )} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} - \frac {2 \, A d^{4}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d^{4}}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {5 \, \sqrt {c x^{2} + b x} B e^{4}}{3 \, c^{3}} - \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {8 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} x}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (B d^{4} + 4 \, A d^{3} e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {4 \, {\left (4 \, B d e^{3} + A e^{4}\right )} x}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {16 \, {\left (B d^{4} + 4 \, A d^{3} e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {{\left (4 \, B d e^{3} + A e^{4}\right )} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {8 \, {\left (B d^{4} + 4 \, A d^{3} e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )}}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {2 \, {\left (4 \, B d e^{3} + A e^{4}\right )} \sqrt {c x^{2} + b x}}{3 \, b c^{2}} + \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )}}{3 \, \sqrt {c x^{2} + b x} b c} \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 )\,{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \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 ) \left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________