3.1215 \(\int (A+B x) (d+e x)^{3/2} (b x+c x^2) \, dx\)

Optimal. Leaf size=126 \[ -\frac {2 (d+e x)^{9/2} (-A c e-b B e+3 B c d)}{9 e^4}+\frac {2 (d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{7 e^4}-\frac {2 d (d+e x)^{5/2} (B d-A e) (c d-b e)}{5 e^4}+\frac {2 B c (d+e x)^{11/2}}{11 e^4} \]

[Out]

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

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Rubi [A]  time = 0.07, 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} \[ -\frac {2 (d+e x)^{9/2} (-A c e-b B e+3 B c d)}{9 e^4}+\frac {2 (d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{7 e^4}-\frac {2 d (d+e x)^{5/2} (B d-A e) (c d-b e)}{5 e^4}+\frac {2 B c (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 114, normalized size = 0.90 \[ \frac {2 (d+e x)^{5/2} \left (11 A e \left (9 b e (5 e x-2 d)+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+B \left (11 b e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 c \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(11*A*e*(9*b*e*(-2*d + 5*e*x) + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) + B*(11*b*e*(8*d^2 - 20*
d*e*x + 35*e^2*x^2) - 3*c*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3))))/(3465*e^4)

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fricas [A]  time = 0.65, size = 190, normalized size = 1.51 \[ \frac {2 \, {\left (315 \, B c e^{5} x^{5} - 48 \, B c d^{5} - 198 \, A b d^{3} e^{2} + 88 \, {\left (B b + A c\right )} d^{4} e + 35 \, {\left (12 \, B c d e^{4} + 11 \, {\left (B b + A c\right )} e^{5}\right )} x^{4} + 5 \, {\left (3 \, B c d^{2} e^{3} + 99 \, A b e^{5} + 110 \, {\left (B b + A c\right )} d e^{4}\right )} x^{3} - 3 \, {\left (6 \, B c d^{3} e^{2} - 264 \, A b d e^{4} - 11 \, {\left (B b + A c\right )} d^{2} e^{3}\right )} x^{2} + {\left (24 \, B c d^{4} e + 99 \, A b d^{2} e^{3} - 44 \, {\left (B b + A c\right )} d^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*B*c*e^5*x^5 - 48*B*c*d^5 - 198*A*b*d^3*e^2 + 88*(B*b + A*c)*d^4*e + 35*(12*B*c*d*e^4 + 11*(B*b + A
*c)*e^5)*x^4 + 5*(3*B*c*d^2*e^3 + 99*A*b*e^5 + 110*(B*b + A*c)*d*e^4)*x^3 - 3*(6*B*c*d^3*e^2 - 264*A*b*d*e^4 -
11*(B*b + A*c)*d^2*e^3)*x^2 + (24*B*c*d^4*e + 99*A*b*d^2*e^3 - 44*(B*b + A*c)*d^3*e^2)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 0.19, size = 667, normalized size = 5.29 \[ \frac {2}{3465} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b d^{2} e^{\left (-1\right )} + 231 \, {\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 )} B b d^{2} e^{\left (-2\right )} + 231 \, {\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 c d^{2} e^{\left (-2\right )} + 99 \, {\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 c d^{2} e^{\left (-3\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 b d e^{\left (-1\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 b d 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 )} A c d e^{\left (-2\right )} + 22 \, {\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 )} B c d e^{\left (-3\right )} + 99 \, {\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 )} A b e^{\left (-1\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 )} B b e^{\left (-2\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 e^{\left (-2\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 e^{\left (-3\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*d^2*e^(-1) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2
)*d + 15*sqrt(x*e + d)*d^2)*B*b*d^2*e^(-2) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*
d^2)*A*c*d^2*e^(-2) + 99*(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*c*d^2*e^(-3) + 462*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b*d*e^(-1) + 19
8*(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*b*d*e^(-2) + 19
8*(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)*A*c*d*e^(-2) + 22
*(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)*B*c*d*e^(-3) + 99*(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)*A*b*e^(-1) + 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)*B*b*e^(-2) + 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*e^(-2) + 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*e^(-3))*e^(-1)

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maple [A]  time = 0.05, size = 121, normalized size = 0.96 \[ -\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (-315 B c \,x^{3} e^{3}-385 A c \,e^{3} x^{2}-385 B b \,e^{3} x^{2}+210 B c d \,e^{2} x^{2}-495 A b \,e^{3} x +220 A c d \,e^{2} x +220 B b d \,e^{2} x -120 B c \,d^{2} e x +198 A b d \,e^{2}-88 A c \,d^{2} e -88 B b \,d^{2} e +48 B c \,d^{3}\right )}{3465 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(e*x+d)^(5/2)*(-315*B*c*e^3*x^3-385*A*c*e^3*x^2-385*B*b*e^3*x^2+210*B*c*d*e^2*x^2-495*A*b*e^3*x+220*A*
c*d*e^2*x+220*B*b*d*e^2*x-120*B*c*d^2*e*x+198*A*b*d*e^2-88*A*c*d^2*e-88*B*b*d^2*e+48*B*c*d^3)/e^4

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maxima [A]  time = 0.49, size = 112, normalized size = 0.89 \[ \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c - 385 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{3465 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*c - 385*(3*B*c*d - (B*b + A*c)*e)*(e*x + d)^(9/2) + 495*(3*B*c*d^2 + A*b*e^2 -
2*(B*b + A*c)*d*e)*(e*x + d)^(7/2) - 693*(B*c*d^3 + A*b*d*e^2 - (B*b + A*c)*d^2*e)*(e*x + d)^(5/2))/e^4

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mupad [B]  time = 0.07, size = 111, normalized size = 0.88 \[ \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{7\,e^4}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{9\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 17.82, size = 434, normalized size = 3.44 \[ \frac {2 A b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {2 A b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {2 A c d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 A c \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 B b d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 B b \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 B c d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {2 B c \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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