3.15.36 \(\int \frac {(A+B x) (a+c x^2)^2}{\sqrt {d+e x}} \, dx\) [1436]

Optimal. Leaf size=216 \[ -\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^6}+\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^6}-\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{5/2}}{5 e^6}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{9/2}}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \]

[Out]

2/3*(a*e^2+c*d^2)*(-4*A*c*d*e+B*a*e^2+5*B*c*d^2)*(e*x+d)^(3/2)/e^6-4/5*c*(-A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2+5*B
*c*d^3)*(e*x+d)^(5/2)/e^6+4/7*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2)*(e*x+d)^(7/2)/e^6-2/9*c^2*(-A*e+5*B*d)*(e*x+d)^
(9/2)/e^6+2/11*B*c^2*(e*x+d)^(11/2)/e^6-2*(-A*e+B*d)*(a*e^2+c*d^2)^2*(e*x+d)^(1/2)/e^6

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Rubi [A]
time = 0.06, antiderivative size = 216, 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 = {786} \begin {gather*} \frac {4 c (d+e x)^{7/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6}-\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6}-\frac {4 c (d+e x)^{5/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}-\frac {2 c^2 (d+e x)^{9/2} (5 B d-A e)}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/Sqrt[d + e*x],x]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)^2*Sqrt[d + e*x])/e^6 + (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*(d
+ e*x)^(3/2))/(3*e^6) - (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(5/2))/(5*e^6) + (4*
c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(7/2))/(7*e^6) - (2*c^2*(5*B*d - A*e)*(d + e*x)^(9/2))/(9*e^6) +
(2*B*c^2*(d + e*x)^(11/2))/(11*e^6)

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 \sqrt {d+e x}}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^5}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^{3/2}}{e^5}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{5/2}}{e^5}+\frac {c^2 (-5 B d+A e) (d+e x)^{7/2}}{e^5}+\frac {B c^2 (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^6}+\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^6}-\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{5/2}}{5 e^6}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{9/2}}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 213, normalized size = 0.99 \begin {gather*} \frac {2 \sqrt {d+e x} \left (11 A e \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (1155 a^2 e^4 (-2 d+e x)+198 a c e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )-5 c^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(11*A*e*(315*a^2*e^4 + 42*a*c*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 - 64*d^3*e*x +
48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)) + B*(1155*a^2*e^4*(-2*d + e*x) + 198*a*c*e^2*(-16*d^3 + 8*d^2*e*
x - 6*d*e^2*x^2 + 5*e^3*x^3) - 5*c^2*(256*d^5 - 128*d^4*e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 -
63*e^5*x^5))))/(3465*e^6)

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Maple [A]
time = 0.68, size = 233, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) c^{2}-4 B \,c^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (A e -B d \right ) c^{2} d +B \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )-4 B \left (e^{2} a +c \,d^{2}\right ) c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right ) c d +B \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{6}}\)	\(233\)
default	\(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) c^{2}-4 B \,c^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (A e -B d \right ) c^{2} d +B \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )-4 B \left (e^{2} a +c \,d^{2}\right ) c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right ) c d +B \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{6}}\)	\(233\)
gosper	\(\frac {2 \sqrt {e x +d}\, \left (315 B \,c^{2} x^{5} e^{5}+385 A \,c^{2} e^{5} x^{4}-350 B \,c^{2} d \,e^{4} x^{4}-440 A \,c^{2} d \,e^{4} x^{3}+990 B a c \,e^{5} x^{3}+400 B \,c^{2} d^{2} e^{3} x^{3}+1386 A a c \,e^{5} x^{2}+528 A \,c^{2} d^{2} e^{3} x^{2}-1188 B a c d \,e^{4} x^{2}-480 B \,c^{2} d^{3} e^{2} x^{2}-1848 A a c d \,e^{4} x -704 A \,c^{2} d^{3} e^{2} x +1155 B \,a^{2} e^{5} x +1584 B a c \,d^{2} e^{3} x +640 B \,c^{2} d^{4} e x +3465 A \,a^{2} e^{5}+3696 A a c \,d^{2} e^{3}+1408 A \,c^{2} d^{4} e -2310 B \,a^{2} d \,e^{4}-3168 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{3465 e^{6}}\)	\(259\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (315 B \,c^{2} x^{5} e^{5}+385 A \,c^{2} e^{5} x^{4}-350 B \,c^{2} d \,e^{4} x^{4}-440 A \,c^{2} d \,e^{4} x^{3}+990 B a c \,e^{5} x^{3}+400 B \,c^{2} d^{2} e^{3} x^{3}+1386 A a c \,e^{5} x^{2}+528 A \,c^{2} d^{2} e^{3} x^{2}-1188 B a c d \,e^{4} x^{2}-480 B \,c^{2} d^{3} e^{2} x^{2}-1848 A a c d \,e^{4} x -704 A \,c^{2} d^{3} e^{2} x +1155 B \,a^{2} e^{5} x +1584 B a c \,d^{2} e^{3} x +640 B \,c^{2} d^{4} e x +3465 A \,a^{2} e^{5}+3696 A a c \,d^{2} e^{3}+1408 A \,c^{2} d^{4} e -2310 B \,a^{2} d \,e^{4}-3168 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{3465 e^{6}}\)	\(259\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (315 B \,c^{2} x^{5} e^{5}+385 A \,c^{2} e^{5} x^{4}-350 B \,c^{2} d \,e^{4} x^{4}-440 A \,c^{2} d \,e^{4} x^{3}+990 B a c \,e^{5} x^{3}+400 B \,c^{2} d^{2} e^{3} x^{3}+1386 A a c \,e^{5} x^{2}+528 A \,c^{2} d^{2} e^{3} x^{2}-1188 B a c d \,e^{4} x^{2}-480 B \,c^{2} d^{3} e^{2} x^{2}-1848 A a c d \,e^{4} x -704 A \,c^{2} d^{3} e^{2} x +1155 B \,a^{2} e^{5} x +1584 B a c \,d^{2} e^{3} x +640 B \,c^{2} d^{4} e x +3465 A \,a^{2} e^{5}+3696 A a c \,d^{2} e^{3}+1408 A \,c^{2} d^{4} e -2310 B \,a^{2} d \,e^{4}-3168 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{3465 e^{6}}\)	\(259\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e^6*(1/11*B*c^2*(e*x+d)^(11/2)+1/9*((A*e-B*d)*c^2-4*B*c^2*d)*(e*x+d)^(9/2)+1/7*(-4*(A*e-B*d)*c^2*d+B*(2*(a*e
^2+c*d^2)*c+4*c^2*d^2))*(e*x+d)^(7/2)+1/5*((A*e-B*d)*(2*(a*e^2+c*d^2)*c+4*c^2*d^2)-4*B*(a*e^2+c*d^2)*c*d)*(e*x
+d)^(5/2)+1/3*(-4*(A*e-B*d)*(a*e^2+c*d^2)*c*d+B*(a*e^2+c*d^2)^2)*(e*x+d)^(3/2)+(A*e-B*d)*(a*e^2+c*d^2)^2*(e*x+
d)^(1/2))

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Maxima [A]
time = 0.28, size = 248, normalized size = 1.15 \begin {gather*} \frac {2}{3465} \, {\left (315 \, {\left (x e + d\right )}^{\frac {11}{2}} B c^{2} - 385 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 990 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 1386 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 3465 \, {\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} \sqrt {x e + d}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(x*e + d)^(11/2)*B*c^2 - 385*(5*B*c^2*d - A*c^2*e)*(x*e + d)^(9/2) + 990*(5*B*c^2*d^2 - 2*A*c^2*d*
e + B*a*c*e^2)*(x*e + d)^(7/2) - 1386*(5*B*c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*(x*e + d)^(5/2
) + 1155*(5*B*c^2*d^4 - 4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*(x*e + d)^(3/2) - 3465*(B
*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)*sqrt(x*e + d))*e^(-6)

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Fricas [A]
time = 3.76, size = 227, normalized size = 1.05 \begin {gather*} -\frac {2}{3465} \, {\left (1280 \, B c^{2} d^{5} - {\left (315 \, B c^{2} x^{5} + 385 \, A c^{2} x^{4} + 990 \, B a c x^{3} + 1386 \, A a c x^{2} + 1155 \, B a^{2} x + 3465 \, A a^{2}\right )} e^{5} + 2 \, {\left (175 \, B c^{2} d x^{4} + 220 \, A c^{2} d x^{3} + 594 \, B a c d x^{2} + 924 \, A a c d x + 1155 \, B a^{2} d\right )} e^{4} - 16 \, {\left (25 \, B c^{2} d^{2} x^{3} + 33 \, A c^{2} d^{2} x^{2} + 99 \, B a c d^{2} x + 231 \, A a c d^{2}\right )} e^{3} + 32 \, {\left (15 \, B c^{2} d^{3} x^{2} + 22 \, A c^{2} d^{3} x + 99 \, B a c d^{3}\right )} e^{2} - 128 \, {\left (5 \, B c^{2} d^{4} x + 11 \, A c^{2} d^{4}\right )} e\right )} \sqrt {x e + d} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/3465*(1280*B*c^2*d^5 - (315*B*c^2*x^5 + 385*A*c^2*x^4 + 990*B*a*c*x^3 + 1386*A*a*c*x^2 + 1155*B*a^2*x + 346
5*A*a^2)*e^5 + 2*(175*B*c^2*d*x^4 + 220*A*c^2*d*x^3 + 594*B*a*c*d*x^2 + 924*A*a*c*d*x + 1155*B*a^2*d)*e^4 - 16
*(25*B*c^2*d^2*x^3 + 33*A*c^2*d^2*x^2 + 99*B*a*c*d^2*x + 231*A*a*c*d^2)*e^3 + 32*(15*B*c^2*d^3*x^2 + 22*A*c^2*
d^3*x + 99*B*a*c*d^3)*e^2 - 128*(5*B*c^2*d^4*x + 11*A*c^2*d^4)*e)*sqrt(x*e + d)*e^(-6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (226) = 452\).
time = 39.48, size = 772, normalized size = 3.57 \begin {gather*} \begin {cases} \frac {- \frac {2 A a^{2} d}{\sqrt {d + e x}} - 2 A a^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {4 A a c d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {4 A a c \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 A c^{2} d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{4}} - \frac {2 A c^{2} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} - \frac {2 B a^{2} d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 B a^{2} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {4 B a c d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{3}} - \frac {4 B a c \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} - \frac {2 B c^{2} d \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{5}} - \frac {2 B c^{2} \left (\frac {d^{6}}{\sqrt {d + e x}} + 6 d^{5} \sqrt {d + e x} - 5 d^{4} \left (d + e x\right )^{\frac {3}{2}} + 4 d^{3} \left (d + e x\right )^{\frac {5}{2}} - \frac {15 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {2 d \left (d + e x\right )^{\frac {9}{2}}}{3} - \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}}}{e} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + \frac {2 A a c x^{3}}{3} + \frac {A c^{2} x^{5}}{5} + \frac {B a^{2} x^{2}}{2} + \frac {B a c x^{4}}{2} + \frac {B c^{2} x^{6}}{6}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*A*a**2*d/sqrt(d + e*x) - 2*A*a**2*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 4*A*a*c*d*(d**2/sqrt(d +
e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 4*A*a*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d
*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 2*A*c**2*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*
(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 2*A*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*
sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/
2)/9)/e**4 - 2*B*a**2*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e - 2*B*a**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x
) - (d + e*x)**(3/2)/3)/e - 4*B*a*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d +
e*x)**(5/2)/5)/e**3 - 4*B*a*c*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d +
e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*B*c**2*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d
+ e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 - 2*B*c**2*(d**
6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)
**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/e, Ne(e, 0)), ((A*a**2*x + 2*A*a*c*x**3/3 + A
*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x**4/2 + B*c**2*x**6/6)/sqrt(d), True))

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Giac [A]
time = 1.18, size = 295, normalized size = 1.37 \begin {gather*} \frac {2}{3465} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{2} e^{\left (-1\right )} + 462 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a c e^{\left (-2\right )} + 198 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a c e^{\left (-3\right )} + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} A c^{2} e^{\left (-4\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} B c^{2} e^{\left (-5\right )} + 3465 \, \sqrt {x e + d} A a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^2*e^(-1) + 462*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*
d + 15*sqrt(x*e + d)*d^2)*A*a*c*e^(-2) + 198*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2 - 35*sqrt(x*e + d)*d^3)*B*a*c*e^(-3) + 11*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*
d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*c^2*e^(-4) + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(
9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5
)*B*c^2*e^(-5) + 3465*sqrt(x*e + d)*A*a^2)*e^(-1)

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Mupad [B]
time = 1.71, size = 197, normalized size = 0.91 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{7\,e^6}+\frac {4\,c\,{\left (d+e\,x\right )}^{5/2}\,\left (-5\,B\,c\,d^3+3\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{5\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,B\,c\,d^2-4\,A\,c\,d\,e+B\,a\,e^2\right )}{3\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,\left (A\,e-B\,d\right )\,\sqrt {d+e\,x}}{e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(7/2)*(20*B*c^2*d^2 + 4*B*a*c*e^2 - 8*A*c^2*d*e))/(7*e^6) + (4*c*(d + e*x)^(5/2)*(A*a*e^3 - 5*B*c*d
^3 - 3*B*a*d*e^2 + 3*A*c*d^2*e))/(5*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11*e^6) + (2*(a*e^2 + c*d^2)*(d + e*x)^
(3/2)*(B*a*e^2 + 5*B*c*d^2 - 4*A*c*d*e))/(3*e^6) + (2*c^2*(A*e - 5*B*d)*(d + e*x)^(9/2))/(9*e^6) + (2*(a*e^2 +
c*d^2)^2*(A*e - B*d)*(d + e*x)^(1/2))/e^6

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