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

Optimal. Leaf size=124 \[ -\frac {2 (d+e x)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac {2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac {2 d \sqrt {d+e x} (B d-A e) (c d-b e)}{e^4}+\frac {2 B c (d+e x)^{7/2}}{7 e^4} \]

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Rubi [A]  time = 0.08, antiderivative size = 124, 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} \begin {gather*} -\frac {2 (d+e x)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac {2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac {2 d \sqrt {d+e x} (B d-A e) (c d-b e)}{e^4}+\frac {2 B c (d+e x)^{7/2}}{7 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.09, size = 113, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {d+e x} \left (7 A e \left (5 b e (e x-2 d)+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+B \left (7 b e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 c \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )\right )}{105 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(7*A*e*(5*b*e*(-2*d + e*x) + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) + B*(7*b*e*(8*d^2 - 4*d*e*x + 3
*e^2*x^2) - 3*c*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3))))/(105*e^4)

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IntegrateAlgebraic [A]  time = 0.08, size = 141, normalized size = 1.14 \begin {gather*} \frac {2 \sqrt {d+e x} \left (35 A b e^2 (d+e x)-105 A b d e^2+105 A c d^2 e-70 A c d e (d+e x)+21 A c e (d+e x)^2+105 b B d^2 e-70 b B d e (d+e x)+21 b B e (d+e x)^2-105 B c d^3+105 B c d^2 (d+e x)-63 B c d (d+e x)^2+15 B c (d+e x)^3\right )}{105 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-105*B*c*d^3 + 105*b*B*d^2*e + 105*A*c*d^2*e - 105*A*b*d*e^2 + 105*B*c*d^2*(d + e*x) - 70*b*
B*d*e*(d + e*x) - 70*A*c*d*e*(d + e*x) + 35*A*b*e^2*(d + e*x) - 63*B*c*d*(d + e*x)^2 + 21*b*B*e*(d + e*x)^2 +
21*A*c*e*(d + e*x)^2 + 15*B*c*(d + e*x)^3))/(105*e^4)

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fricas [A]  time = 0.40, size = 108, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (15 \, B c e^{3} x^{3} - 48 \, B c d^{3} - 70 \, A b d e^{2} + 56 \, {\left (B b + A c\right )} d^{2} e - 3 \, {\left (6 \, B c d e^{2} - 7 \, {\left (B b + A c\right )} e^{3}\right )} x^{2} + {\left (24 \, B c d^{2} e + 35 \, A b e^{3} - 28 \, {\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*B*c*e^3*x^3 - 48*B*c*d^3 - 70*A*b*d*e^2 + 56*(B*b + A*c)*d^2*e - 3*(6*B*c*d*e^2 - 7*(B*b + A*c)*e^3)
*x^2 + (24*B*c*d^2*e + 35*A*b*e^3 - 28*(B*b + A*c)*d*e^2)*x)*sqrt(e*x + d)/e^4

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giac [A]  time = 0.18, size = 167, normalized size = 1.35 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b e^{\left (-1\right )} + 7 \, {\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 e^{\left (-2\right )} + 7 \, {\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 e^{\left (-2\right )} + 3 \, {\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 e^{\left (-3\right )}\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+b*x)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*e^(-1) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*
sqrt(x*e + d)*d^2)*B*b*e^(-2) + 7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*c*e^(-2)
+ 3*(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*e^(-3))*e^
(-1)

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maple [A]  time = 0.05, size = 121, normalized size = 0.98 \begin {gather*} -\frac {2 \left (-15 B c \,x^{3} e^{3}-21 A c \,e^{3} x^{2}-21 B b \,e^{3} x^{2}+18 B c d \,e^{2} x^{2}-35 A b \,e^{3} x +28 A c d \,e^{2} x +28 B b d \,e^{2} x -24 B c \,d^{2} e x +70 A b d \,e^{2}-56 A c \,d^{2} e -56 B b \,d^{2} e +48 B c \,d^{3}\right ) \sqrt {e x +d}}{105 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(-15*B*c*e^3*x^3-21*A*c*e^3*x^2-21*B*b*e^3*x^2+18*B*c*d*e^2*x^2-35*A*b*e^3*x+28*A*c*d*e^2*x+28*B*b*d*e^
2*x-24*B*c*d^2*e*x+70*A*b*d*e^2-56*A*c*d^2*e-56*B*b*d^2*e+48*B*c*d^3)*(e*x+d)^(1/2)/e^4

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maxima [A]  time = 0.60, size = 112, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c - 21 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\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 {3}{2}} - 105 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \sqrt {e x + d}\right )}}{105 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(e*x + d)^(7/2)*B*c - 21*(3*B*c*d - (B*b + A*c)*e)*(e*x + d)^(5/2) + 35*(3*B*c*d^2 + A*b*e^2 - 2*(B*
b + A*c)*d*e)*(e*x + d)^(3/2) - 105*(B*c*d^3 + A*b*d*e^2 - (B*b + A*c)*d^2*e)*sqrt(e*x + d))/e^4

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mupad [B]  time = 0.07, size = 111, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{3\,e^4}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{5\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}}{e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 47.60, size = 430, normalized size = 3.47 \begin {gather*} \begin {cases} \frac {- \frac {2 A b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 A b \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 {2 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 {2 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 B b 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 {2 B b \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 B 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 {2 B 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}}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {A b x^{2}}{2} + \frac {B c x^{4}}{4} + \frac {x^{3} \left (A c + B b\right )}{3}}{\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+b*x)/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*A*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e - 2*A*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d
+ e*x)**(3/2)/3)/e - 2*A*c*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 2*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*B*b*d*(d**2/sqrt(d
+ e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 2*B*b*(-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*B*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 - 2*B*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)/e, Ne(e, 0)), ((A*b*x**2/2 + B*c*x**4/4 + x**3*(A*c +
B*b)/3)/sqrt(d), True))

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