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

Optimal. Leaf size=116 \[ \frac {2 (d+e x)^{7/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac {2 c (d+e x)^{9/2} (3 B d-A e)}{9 e^4}+\frac {2 B c (d+e x)^{11/2}}{11 e^4} \]

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Rubi [A]  time = 0.06, antiderivative size = 116, 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 (d+e x)^{7/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac {2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac {2 c (d+e x)^{9/2} (3 B d-A e)}{9 e^4}+\frac {2 B c (d+e x)^{11/2}}{11 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^(5/2))/(5*e^4) + (2*(3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(7/2
))/(7*e^4) - (2*c*(3*B*d - A*e)*(d + e*x)^(9/2))/(9*e^4) + (2*B*c*(d + e*x)^(11/2))/(11*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 (A+B x) (d+e x)^{3/2} \left (a+c x^2\right ) \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right ) (d+e x)^{3/2}}{e^3}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{5/2}}{e^3}+\frac {c (-3 B d+A e) (d+e x)^{7/2}}{e^3}+\frac {B c (d+e x)^{9/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (B d-A e) \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^4}+\frac {2 \left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{7/2}}{7 e^4}-\frac {2 c (3 B d-A 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 = 0.09, size = 99, normalized size = 0.85 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (11 A e \left (63 a e^2+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-3 B \left (33 a e^2 (2 d-5 e x)+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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(11*A*e*(63*a*e^2 + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 3*B*(33*a*e^2*(2*d - 5*e*x) + c*(1
6*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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IntegrateAlgebraic [A]  time = 0.08, size = 117, normalized size = 1.01 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (693 a A e^3+495 a B e^2 (d+e x)-693 a B d e^2+693 A c d^2 e-990 A c d e (d+e x)+385 A c e (d+e x)^2-693 B c d^3+1485 B c d^2 (d+e x)-1155 B c d (d+e x)^2+315 B c (d+e x)^3\right )}{3465 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(-693*B*c*d^3 + 693*A*c*d^2*e - 693*a*B*d*e^2 + 693*a*A*e^3 + 1485*B*c*d^2*(d + e*x) - 990*
A*c*d*e*(d + e*x) + 495*a*B*e^2*(d + e*x) - 1155*B*c*d*(d + e*x)^2 + 385*A*c*e*(d + e*x)^2 + 315*B*c*(d + e*x)
^3))/(3465*e^4)

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fricas [A]  time = 0.41, size = 190, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (315 \, B c e^{5} x^{5} - 48 \, B c d^{5} + 88 \, A c d^{4} e - 198 \, B a d^{3} e^{2} + 693 \, A a d^{2} e^{3} + 35 \, {\left (12 \, B c d e^{4} + 11 \, A c e^{5}\right )} x^{4} + 5 \, {\left (3 \, B c d^{2} e^{3} + 110 \, A c d e^{4} + 99 \, B a e^{5}\right )} x^{3} - 3 \, {\left (6 \, B c d^{3} e^{2} - 11 \, A c d^{2} e^{3} - 264 \, B a d e^{4} - 231 \, A a e^{5}\right )} x^{2} + {\left (24 \, B c d^{4} e - 44 \, A c d^{3} e^{2} + 99 \, B a d^{2} e^{3} + 1386 \, A a d e^{4}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*B*c*e^5*x^5 - 48*B*c*d^5 + 88*A*c*d^4*e - 198*B*a*d^3*e^2 + 693*A*a*d^2*e^3 + 35*(12*B*c*d*e^4 + 1
1*A*c*e^5)*x^4 + 5*(3*B*c*d^2*e^3 + 110*A*c*d*e^4 + 99*B*a*e^5)*x^3 - 3*(6*B*c*d^3*e^2 - 11*A*c*d^2*e^3 - 264*
B*a*d*e^4 - 231*A*a*e^5)*x^2 + (24*B*c*d^4*e - 44*A*c*d^3*e^2 + 99*B*a*d^2*e^3 + 1386*A*a*d*e^4)*x)*sqrt(e*x +
d)/e^4

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giac [B]  time = 0.35, size = 580, normalized size = 5.00 \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 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 )} 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 )} B a 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 )} 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 )} + 3465 \, \sqrt {x e + d} A a d^{2} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a d + 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 a 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 )} 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 )} + 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 a\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*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)*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)*B*a*d*e^(-1) + 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)*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) + 3465*sqrt(x*e + d)*A*a*d^2 + 2310*((x*e + d)^(3/2) - 3*sqrt(
x*e + d)*d)*A*a*d + 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*a*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)*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) + 231*(3*(x*e
+ d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a)*e^(-1)

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maple [A]  time = 0.05, size = 101, normalized size = 0.87 \begin {gather*} \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}-210 B c d \,e^{2} x^{2}-220 A c d \,e^{2} x +495 B a \,e^{3} x +120 B c \,d^{2} e x +693 a A \,e^{3}+88 A c \,d^{2} e -198 a B d \,e^{2}-48 B c \,d^{3}\right )}{3465 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(e*x+d)^(5/2)*(315*B*c*e^3*x^3+385*A*c*e^3*x^2-210*B*c*d*e^2*x^2-220*A*c*d*e^2*x+495*B*a*e^3*x+120*B*c*
d^2*e*x+693*A*a*e^3+88*A*c*d^2*e-198*B*a*d*e^2-48*B*c*d^3)/e^4

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maxima [A]  time = 0.64, size = 104, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c - 385 \, {\left (3 \, B c d - A c e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{3465 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 100, normalized size = 0.86 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,B\,c\,d^2-4\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{7\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,c\,\left (A\,e-3\,B\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,\left (c\,d^2+a\,e^2\right )\,\left (A\,e-B\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,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)^(7/2)*(2*B*a*e^2 + 6*B*c*d^2 - 4*A*c*d*e))/(7*e^4) + (2*B*c*(d + e*x)^(11/2))/(11*e^4) + (2*c*(A*e
- 3*B*d)*(d + e*x)^(9/2))/(9*e^4) + (2*(a*e^2 + c*d^2)*(A*e - B*d)*(d + e*x)^(5/2))/(5*e^4)

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sympy [A]  time = 15.78, size = 379, normalized size = 3.27 \begin {gather*} A a d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 A a \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \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 a 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 B a \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 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a*(-d*(d + e*x)**(3/2)/3 + (d +
e*x)**(5/2)/5)/e + 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*a*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*B*a*(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*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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