3.13.58 \(\int \frac {(A+B x) (a+c x^2)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=112 \[ \frac {2 \sqrt {d+e x} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{e^4 \sqrt {d+e x}}-\frac {2 c (d+e x)^{3/2} (3 B d-A e)}{3 e^4}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \begin {gather*} \frac {2 \sqrt {d+e x} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{e^4 \sqrt {d+e x}}-\frac {2 c (d+e x)^{3/2} (3 B d-A e)}{3 e^4}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/(d + e*x)^(3/2),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 97, normalized size = 0.87 \begin {gather*} \frac {6 B \left (5 a e^2 (2 d+e x)+c \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-10 A e \left (3 a e^2+c \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^(3/2),x]

[Out]

(-10*A*e*(3*a*e^2 + c*(8*d^2 + 4*d*e*x - e^2*x^2)) + 6*B*(5*a*e^2*(2*d + e*x) + c*(16*d^3 + 8*d^2*e*x - 2*d*e^
2*x^2 + e^3*x^3)))/(15*e^4*Sqrt[d + e*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.07, size = 117, normalized size = 1.04 \begin {gather*} \frac {2 \left (-15 a A e^3+15 a B e^2 (d+e x)+15 a B d e^2-15 A c d^2 e-30 A c d e (d+e x)+5 A c e (d+e x)^2+15 B c d^3+45 B c d^2 (d+e x)-15 B c d (d+e x)^2+3 B c (d+e x)^3\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2))/(d + e*x)^(3/2),x]

[Out]

(2*(15*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 15*a*A*e^3 + 45*B*c*d^2*(d + e*x) - 30*A*c*d*e*(d + e*x) + 15*a
*B*e^2*(d + e*x) - 15*B*c*d*(d + e*x)^2 + 5*A*c*e*(d + e*x)^2 + 3*B*c*(d + e*x)^3))/(15*e^4*Sqrt[d + e*x])

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 110, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 40 \, A c d^{2} e + 30 \, B a d e^{2} - 15 \, A a e^{3} - {\left (6 \, B c d e^{2} - 5 \, A c e^{3}\right )} x^{2} + {\left (24 \, B c d^{2} e - 20 \, A c d e^{2} + 15 \, B a e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{5} x + d e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*B*c*e^3*x^3 + 48*B*c*d^3 - 40*A*c*d^2*e + 30*B*a*d*e^2 - 15*A*a*e^3 - (6*B*c*d*e^2 - 5*A*c*e^3)*x^2 +
(24*B*c*d^2*e - 20*A*c*d*e^2 + 15*B*a*e^3)*x)*sqrt(e*x + d)/(e^5*x + d*e^4)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 135, normalized size = 1.21 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B c e^{16} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B c d e^{16} + 45 \, \sqrt {x e + d} B c d^{2} e^{16} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c e^{17} - 30 \, \sqrt {x e + d} A c d e^{17} + 15 \, \sqrt {x e + d} B a e^{18}\right )} e^{\left (-20\right )} + \frac {2 \, {\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} e^{\left (-4\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*c*e^16 - 15*(x*e + d)^(3/2)*B*c*d*e^16 + 45*sqrt(x*e + d)*B*c*d^2*e^16 + 5*(x*e + d)
^(3/2)*A*c*e^17 - 30*sqrt(x*e + d)*A*c*d*e^17 + 15*sqrt(x*e + d)*B*a*e^18)*e^(-20) + 2*(B*c*d^3 - A*c*d^2*e +
B*a*d*e^2 - A*a*e^3)*e^(-4)/sqrt(x*e + d)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 101, normalized size = 0.90 \begin {gather*} -\frac {2 \left (-3 B c \,x^{3} e^{3}-5 A c \,e^{3} x^{2}+6 B c d \,e^{2} x^{2}+20 A c d \,e^{2} x -15 B a \,e^{3} x -24 B c \,d^{2} e x +15 a A \,e^{3}+40 A c \,d^{2} e -30 a B d \,e^{2}-48 B c \,d^{3}\right )}{15 \sqrt {e x +d}\, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)/(e*x+d)^(3/2),x)

[Out]

-2/15/(e*x+d)^(1/2)*(-3*B*c*e^3*x^3-5*A*c*e^3*x^2+6*B*c*d*e^2*x^2+20*A*c*d*e^2*x-15*B*a*e^3*x-24*B*c*d^2*e*x+1
5*A*a*e^3+40*A*c*d^2*e-30*B*a*d*e^2-48*B*c*d^3)/e^4

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 112, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c - 5 \, {\left (3 \, B c d - A c e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} \sqrt {e x + d}}{e^{3}} + \frac {15 \, {\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )}}{\sqrt {e x + d} e^{3}}\right )}}{15 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/15*((3*(e*x + d)^(5/2)*B*c - 5*(3*B*c*d - A*c*e)*(e*x + d)^(3/2) + 15*(3*B*c*d^2 - 2*A*c*d*e + B*a*e^2)*sqrt
(e*x + d))/e^3 + 15*(B*c*d^3 - A*c*d^2*e + B*a*d*e^2 - A*a*e^3)/(sqrt(e*x + d)*e^3))/e

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 111, normalized size = 0.99 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (6\,B\,c\,d^2-4\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{e^4}-\frac {-2\,B\,c\,d^3+2\,A\,c\,d^2\,e-2\,B\,a\,d\,e^2+2\,A\,a\,e^3}{e^4\,\sqrt {d+e\,x}}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,c\,\left (A\,e-3\,B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)*(A + B*x))/(d + e*x)^(3/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 21.21, size = 112, normalized size = 1.00 \begin {gather*} \frac {2 B c \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (2 A c e - 6 B c d\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (- 4 A c d e + 2 B a e^{2} + 6 B c d^{2}\right )}{e^{4}} + \frac {2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )}{e^{4} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)/(e*x+d)**(3/2),x)

[Out]

2*B*c*(d + e*x)**(5/2)/(5*e**4) + (d + e*x)**(3/2)*(2*A*c*e - 6*B*c*d)/(3*e**4) + sqrt(d + e*x)*(-4*A*c*d*e +
2*B*a*e**2 + 6*B*c*d**2)/e**4 + 2*(-A*e + B*d)*(a*e**2 + c*d**2)/(e**4*sqrt(d + e*x))

________________________________________________________________________________________