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

Optimal. Leaf size=114 \[ \frac {2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4}-\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right ) (B d-A e)}{e^4}-\frac {2 c (d+e x)^{5/2} (3 B d-A e)}{5 e^4}+\frac {2 B c (d+e x)^{7/2}}{7 e^4} \]

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Rubi [A]  time = 0.05, antiderivative size = 114, 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)^{3/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4}-\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right ) (B d-A e)}{e^4}-\frac {2 c (d+e x)^{5/2} (3 B d-A e)}{5 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)*(a + c*x^2))/Sqrt[d + e*x],x]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^4 + (2*(3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/(3*
e^4) - (2*c*(3*B*d - A*e)*(d + e*x)^(5/2))/(5*e^4) + (2*B*c*(d + e*x)^(7/2))/(7*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 )}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )}{e^3 \sqrt {d+e x}}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^3}+\frac {c (-3 B d+A e) (d+e x)^{3/2}}{e^3}+\frac {B c (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (B d-A e) \left (c d^2+a e^2\right ) \sqrt {d+e x}}{e^4}+\frac {2 \left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^4}-\frac {2 c (3 B d-A 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 = 0.08, size = 96, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {d+e x} \left (105 a A e^3+35 a B e^2 (e x-2 d)+7 A c e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 B c \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )}{105 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(105*a*A*e^3 + 35*a*B*e^2*(-2*d + e*x) + 7*A*c*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) - 3*B*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 = 0.07, size = 117, normalized size = 1.03 \begin {gather*} \frac {2 \sqrt {d+e x} \left (105 a A e^3+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 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)*(a + c*x^2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(-105*B*c*d^3 + 105*A*c*d^2*e - 105*a*B*d*e^2 + 105*a*A*e^3 + 105*B*c*d^2*(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*A*c*e*(d + e*x)^2 + 15*B*c*(d + e*x)^3))/(105
*e^4)

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fricas [A]  time = 0.43, size = 100, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (15 \, B c e^{3} x^{3} - 48 \, B c d^{3} + 56 \, A c d^{2} e - 70 \, B a d e^{2} + 105 \, A a e^{3} - 3 \, {\left (6 \, B c d e^{2} - 7 \, A c e^{3}\right )} x^{2} + {\left (24 \, B c d^{2} e - 28 \, A c d e^{2} + 35 \, B a e^{3}\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+a)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 138, normalized size = 1.21 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a 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 )} 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 )} + 105 \, \sqrt {x e + d} A a\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+a)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*e^(-1) + 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) + 105*sqrt(x*e + d)*A*a)*e^(-1)

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maple [A]  time = 0.05, size = 101, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (15 B c \,x^{3} e^{3}+21 A c \,e^{3} x^{2}-18 B c d \,e^{2} x^{2}-28 A c d \,e^{2} x +35 B a \,e^{3} x +24 B c \,d^{2} e x +105 a A \,e^{3}+56 A c \,d^{2} e -70 a B d \,e^{2}-48 B c \,d^{3}\right )}{105 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)^(1/2),x)

[Out]

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

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maxima [A]  time = 0.58, size = 104, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c - 21 \, {\left (3 \, B c d - A c e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 105 \, {\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\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+a)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.07, size = 100, normalized size = 0.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,B\,c\,d^2-4\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{3\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,c\,\left (A\,e-3\,B\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,\left (c\,d^2+a\,e^2\right )\,\left (A\,e-B\,d\right )\,\sqrt {d+e\,x}}{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)^(1/2),x)

[Out]

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

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sympy [A]  time = 36.49, size = 374, normalized size = 3.28 \begin {gather*} \begin {cases} \frac {- \frac {2 A a d}{\sqrt {d + e x}} - 2 A a \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \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 a d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 B a \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 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 {A a x + \frac {A c x^{3}}{3} + \frac {B a x^{2}}{2} + \frac {B c x^{4}}{4}}{\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+a)/(e*x+d)**(1/2),x)

[Out]

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

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