3.1436 \(\int \frac{(A+B x) (a+c x^2)^2}{\sqrt{d+e x}} \, dx\)

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

[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)

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Rubi [A]  time = 0.0948489, antiderivative size = 216, normalized size of antiderivative = 1., 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 = {772} \[ \frac{4 c (d+e x)^{7/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}-\frac{4 c (d+e x)^{5/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6}-\frac{2 c^2 (d+e x)^{9/2} (5 B d-A e)}{9 e^6}+\frac{2 B c^2 (d+e x)^{11/2}}{11 e^6} \]

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 772

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*}

Mathematica [A]  time = 0.171641, size = 213, normalized size = 0.99 \[ \frac{2 \sqrt{d+e x} \left (11 A e \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (1155 a^2 e^4 (e x-2 d)+198 a c e^2 \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )-5 c^2 \left (96 d^3 e^2 x^2-80 d^2 e^3 x^3-128 d^4 e x+256 d^5+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \]

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 = 0.007, size = 259, normalized size = 1.2 \begin{align*}{\frac{630\,B{c}^{2}{x}^{5}{e}^{5}+770\,A{c}^{2}{e}^{5}{x}^{4}-700\,B{c}^{2}d{e}^{4}{x}^{4}-880\,A{c}^{2}d{e}^{4}{x}^{3}+1980\,Bac{e}^{5}{x}^{3}+800\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+2772\,Aac{e}^{5}{x}^{2}+1056\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-2376\,Bacd{e}^{4}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-3696\,Aacd{e}^{4}x-1408\,A{c}^{2}{d}^{3}{e}^{2}x+2310\,B{a}^{2}{e}^{5}x+3168\,Bac{d}^{2}{e}^{3}x+1280\,B{c}^{2}{d}^{4}ex+6930\,A{a}^{2}{e}^{5}+7392\,A{d}^{2}ac{e}^{3}+2816\,A{d}^{4}{c}^{2}e-4620\,B{a}^{2}d{e}^{4}-6336\,aBc{d}^{3}{e}^{2}-2560\,B{c}^{2}{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \end{align*}

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)

[Out]

2/3465*(e*x+d)^(1/2)*(315*B*c^2*e^5*x^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)/e^6

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Maxima [A]  time = 0.99946, size = 335, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c^{2} - 385 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + 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 (e x + 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 (e x + 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 (e x + 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{e x + d}\right )}}{3465 \, e^{6}} \end{align*}

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*(e*x + d)^(11/2)*B*c^2 - 385*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(9/2) + 990*(5*B*c^2*d^2 - 2*A*c^2*d*
e + B*a*c*e^2)*(e*x + 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)*(e*x + 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)*(e*x + 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(e*x + d))/e^6

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Fricas [A]  time = 1.54182, size = 594, normalized size = 2.75 \begin{align*} \frac{2 \,{\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1408 \, A c^{2} d^{4} e - 3168 \, B a c d^{3} e^{2} + 3696 \, A a c d^{2} e^{3} - 2310 \, B a^{2} d e^{4} + 3465 \, A a^{2} e^{5} - 35 \,{\left (10 \, B c^{2} d e^{4} - 11 \, A c^{2} e^{5}\right )} x^{4} + 10 \,{\left (40 \, B c^{2} d^{2} e^{3} - 44 \, A c^{2} d e^{4} + 99 \, B a c e^{5}\right )} x^{3} - 6 \,{\left (80 \, B c^{2} d^{3} e^{2} - 88 \, A c^{2} d^{2} e^{3} + 198 \, B a c d e^{4} - 231 \, A a c e^{5}\right )} x^{2} +{\left (640 \, B c^{2} d^{4} e - 704 \, A c^{2} d^{3} e^{2} + 1584 \, B a c d^{2} e^{3} - 1848 \, A a c d e^{4} + 1155 \, B a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \end{align*}

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*(315*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 1408*A*c^2*d^4*e - 3168*B*a*c*d^3*e^2 + 3696*A*a*c*d^2*e^3 - 2310
*B*a^2*d*e^4 + 3465*A*a^2*e^5 - 35*(10*B*c^2*d*e^4 - 11*A*c^2*e^5)*x^4 + 10*(40*B*c^2*d^2*e^3 - 44*A*c^2*d*e^4
+ 99*B*a*c*e^5)*x^3 - 6*(80*B*c^2*d^3*e^2 - 88*A*c^2*d^2*e^3 + 198*B*a*c*d*e^4 - 231*A*a*c*e^5)*x^2 + (640*B*
c^2*d^4*e - 704*A*c^2*d^3*e^2 + 1584*B*a*c*d^2*e^3 - 1848*A*a*c*d*e^4 + 1155*B*a^2*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 59.9598, size = 772, normalized size = 3.57 \begin{align*} \text{result too large to display} \end{align*}

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 = 1.14016, size = 398, normalized size = 1.84 \begin{align*} \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{align*}

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)