3.21.39 \(\int \frac {(d+e x)^{7/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=292 \[ \frac {16 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d+e x}}+\frac {8 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^3 e^2}+\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{5 c^2 e^2 (2 c d-b e)}+\frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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Rubi [A]  time = 0.41, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {788, 656, 648} \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{5 c^2 e^2 (2 c d-b e)}+\frac {8 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^3 e^2}+\frac {16 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d+e x}}+\frac {2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(7/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(7/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(16*(2*c*d - b*e)*(5*c*e*f + 7*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*c^4*e^2*Sqrt[d
+ e*x]) + (8*(5*c*e*f + 7*c*d*g - 6*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*c^3*e^
2) + (2*(5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(5*c^2*e^2*(2
*c*d - b*e))

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 788

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(5 c e f+7 c d g-6 b e g) \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}-\frac {(4 (5 c e f+7 c d g-6 b e g)) \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{5 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 (5 c e f+7 c d g-6 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^3 e^2}+\frac {2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}-\frac {(8 (2 c d-b e) (5 c e f+7 c d g-6 b e g)) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{15 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (2 c d-b e) (5 c e f+7 c d g-6 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^4 e^2 \sqrt {d+e x}}+\frac {8 (5 c e f+7 c d g-6 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^3 e^2}+\frac {2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 168, normalized size = 0.58 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (48 b^3 e^3 g-8 b^2 c e^2 (28 d g+5 e f-3 e g x)+2 b c^2 e \left (167 d^2 g+d e (70 f-44 g x)-e^2 x (10 f+3 g x)\right )+c^3 \left (-158 d^3 g+d^2 e (79 g x-115 f)+2 d e^2 x (25 f+8 g x)+e^3 x^2 (5 f+3 g x)\right )\right )}{15 c^4 e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(7/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(48*b^3*e^3*g - 8*b^2*c*e^2*(5*e*f + 28*d*g - 3*e*g*x) + 2*b*c^2*e*(167*d^2*g + d*e*(70*f -
44*g*x) - e^2*x*(10*f + 3*g*x)) + c^3*(-158*d^3*g + e^3*x^2*(5*f + 3*g*x) + 2*d*e^2*x*(25*f + 8*g*x) + d^2*e*(
-115*f + 79*g*x))))/(15*c^4*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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IntegrateAlgebraic [A]  time = 5.15, size = 265, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (48 b^3 e^3 g+24 b^2 c e^2 g (d+e x)-248 b^2 c d e^2 g-40 b^2 c e^3 f+416 b c^2 d^2 e g-20 b c^2 e^2 f (d+e x)+160 b c^2 d e^2 f-6 b c^2 e g (d+e x)^2-76 b c^2 d e g (d+e x)-224 c^3 d^3 g-160 c^3 d^2 e f+56 c^3 d^2 g (d+e x)+5 c^3 e f (d+e x)^2+40 c^3 d e f (d+e x)+3 c^3 g (d+e x)^3+7 c^3 d g (d+e x)^2\right )}{15 c^4 e^2 \sqrt {d+e x} (b e+c (d+e x)-2 c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((d + e*x)^(7/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2]*(-160*c^3*d^2*e*f + 160*b*c^2*d*e^2*f - 40*b^2*c*e^3*f - 224*
c^3*d^3*g + 416*b*c^2*d^2*e*g - 248*b^2*c*d*e^2*g + 48*b^3*e^3*g + 40*c^3*d*e*f*(d + e*x) - 20*b*c^2*e^2*f*(d
+ e*x) + 56*c^3*d^2*g*(d + e*x) - 76*b*c^2*d*e*g*(d + e*x) + 24*b^2*c*e^2*g*(d + e*x) + 5*c^3*e*f*(d + e*x)^2
+ 7*c^3*d*g*(d + e*x)^2 - 6*b*c^2*e*g*(d + e*x)^2 + 3*c^3*g*(d + e*x)^3))/(15*c^4*e^2*Sqrt[d + e*x]*(-2*c*d +
b*e + c*(d + e*x)))

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fricas [A]  time = 0.41, size = 257, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (3 \, c^{3} e^{3} g x^{3} + {\left (5 \, c^{3} e^{3} f + 2 \, {\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} - 5 \, {\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f - 2 \, {\left (79 \, c^{3} d^{3} - 167 \, b c^{2} d^{2} e + 112 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g + {\left (10 \, {\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (c^{5} e^{4} x^{2} + b c^{4} e^{4} x - c^{5} d^{2} e^{2} + b c^{4} d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*c^3*e^3*g*x^3 + (5*c^3*e^3*f + 2*(8*c^3*d*e^2 - 3*b*c^2*e^3)*g)*x^2 - 5*(23*c^3*d^2*e - 28*b*c^2*d*e^2
+ 8*b^2*c*e^3)*f - 2*(79*c^3*d^3 - 167*b*c^2*d^2*e + 112*b^2*c*d*e^2 - 24*b^3*e^3)*g + (10*(5*c^3*d*e^2 - 2*b
*c^2*e^3)*f + (79*c^3*d^2*e - 88*b*c^2*d*e^2 + 24*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*
sqrt(e*x + d)/(c^5*e^4*x^2 + b*c^4*e^4*x - c^5*d^2*e^2 + b*c^4*d*e^3)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [b,c,d,exp(1),exp(2)]=[-8,-96,87,35,14]Warning, need to choose a branch for the root of a polyno
mial with parameters. This might be wrong.The choice was done assuming [b,c,d,exp(1),exp(2)]=[57,18,-10,85,-42
]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice
was done assuming [b,c,d,exp(1),exp(2)]=[-23,67,97,57,86]Warning, need to choose a branch for the root of a p
olynomial with parameters. This might be wrong.The choice was done assuming [b,c,d,exp(1),exp(2)]=[45,19,-66,-
61,-5]Evaluation time: 121.97Unable to transpose Error: Bad Argument Value

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maple [A]  time = 0.05, size = 235, normalized size = 0.80 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (3 g \,e^{3} x^{3} c^{3}-6 b \,c^{2} e^{3} g \,x^{2}+16 c^{3} d \,e^{2} g \,x^{2}+5 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -88 b \,c^{2} d \,e^{2} g x -20 b \,c^{2} e^{3} f x +79 c^{3} d^{2} e g x +50 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -224 b^{2} c d \,e^{2} g -40 b^{2} c \,e^{3} f +334 b \,c^{2} d^{2} e g +140 b \,c^{2} d \,e^{2} f -158 c^{3} d^{3} g -115 f \,d^{2} c^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{15 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} c^{4} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

2/15*(c*e*x+b*e-c*d)*(3*c^3*e^3*g*x^3-6*b*c^2*e^3*g*x^2+16*c^3*d*e^2*g*x^2+5*c^3*e^3*f*x^2+24*b^2*c*e^3*g*x-88
*b*c^2*d*e^2*g*x-20*b*c^2*e^3*f*x+79*c^3*d^2*e*g*x+50*c^3*d*e^2*f*x+48*b^3*e^3*g-224*b^2*c*d*e^2*g-40*b^2*c*e^
3*f+334*b*c^2*d^2*e*g+140*b*c^2*d*e^2*f-158*c^3*d^3*g-115*c^3*d^2*e*f)*(e*x+d)^(3/2)/c^4/e^2/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(3/2)

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maxima [A]  time = 0.73, size = 203, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (c^{2} e^{2} x^{2} - 23 \, c^{2} d^{2} + 28 \, b c d e - 8 \, b^{2} e^{2} + 2 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, \sqrt {-c e x + c d - b e} c^{3} e} - \frac {2 \, {\left (3 \, c^{3} e^{3} x^{3} - 158 \, c^{3} d^{3} + 334 \, b c^{2} d^{2} e - 224 \, b^{2} c d e^{2} + 48 \, b^{3} e^{3} + 2 \, {\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} x^{2} + {\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} x\right )} g}{15 \, \sqrt {-c e x + c d - b e} c^{4} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/3*(c^2*e^2*x^2 - 23*c^2*d^2 + 28*b*c*d*e - 8*b^2*e^2 + 2*(5*c^2*d*e - 2*b*c*e^2)*x)*f/(sqrt(-c*e*x + c*d -
b*e)*c^3*e) - 2/15*(3*c^3*e^3*x^3 - 158*c^3*d^3 + 334*b*c^2*d^2*e - 224*b^2*c*d*e^2 + 48*b^3*e^3 + 2*(8*c^3*d*
e^2 - 3*b*c^2*e^3)*x^2 + (79*c^3*d^2*e - 88*b*c^2*d*e^2 + 24*b^2*c*e^3)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^4*e^2
)

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mupad [B]  time = 2.95, size = 267, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (16\,c\,d\,g-6\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^3\,e^2}-\frac {\sqrt {d+e\,x}\,\left (-96\,g\,b^3\,e^3+448\,g\,b^2\,c\,d\,e^2+80\,f\,b^2\,c\,e^3-668\,g\,b\,c^2\,d^2\,e-280\,f\,b\,c^2\,d\,e^2+316\,g\,c^3\,d^3+230\,f\,c^3\,d^2\,e\right )}{15\,c^5\,e^4}+\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{5\,c^2\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-176\,g\,b\,c^2\,d\,e^2-40\,f\,b\,c^2\,e^3+158\,g\,c^3\,d^2\,e+100\,f\,c^3\,d\,e^2\right )}{15\,c^5\,e^4}\right )}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(d + e*x)^(7/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),x)

[Out]

((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*x^2*(d + e*x)^(1/2)*(16*c*d*g - 6*b*e*g + 5*c*e*f))/(15*c^3*e
^2) - ((d + e*x)^(1/2)*(316*c^3*d^3*g - 96*b^3*e^3*g + 80*b^2*c*e^3*f + 230*c^3*d^2*e*f - 280*b*c^2*d*e^2*f -
668*b*c^2*d^2*e*g + 448*b^2*c*d*e^2*g))/(15*c^5*e^4) + (2*g*x^3*(d + e*x)^(1/2))/(5*c^2*e) + (x*(d + e*x)^(1/2
)*(48*b^2*c*e^3*g - 40*b*c^2*e^3*f + 100*c^3*d*e^2*f + 158*c^3*d^2*e*g - 176*b*c^2*d*e^2*g))/(15*c^5*e^4)))/(x
^2 + (b*x)/c + (d*(b*e - c*d))/(c*e^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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