3.2492 \(\int \frac{(A+B x) (d+e x)}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=225 \[ -\frac{16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]

[Out]

(2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(5*c*(b^2 -
4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b + 2*c*x))/(15*
c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b +
2*c*x))/(15*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.183504, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {777, 614, 613} \[ -\frac{16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(5*c*(b^2 -
4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b + 2*c*x))/(15*
c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b +
2*c*x))/(15*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{\left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{\left (8 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2}\\ &=\frac{2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.69248, size = 200, normalized size = 0.89 \[ \frac{2 \left (-3 \left (b^2-4 a c\right )^2 (A c (-2 a e+b (d-e x)+2 c d x)+B (a b e-2 a c (d+e x)+b x (b e-c d)))+\left (b^2-4 a c\right ) (b+2 c x) (a+x (b+c x)) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )-8 c (b+2 c x) (a+x (b+c x))^2 \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )\right )}{15 c \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*((b^2 - 4*a*c)*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b + 2*c*x)*(a + x*(b + c*x)) - 8*c*
(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b + 2*c*x)*(a + x*(b + c*x))^2 - 3*(b^2 - 4*a*c)^2*(A
*c*(-2*a*e + 2*c*d*x + b*(d - e*x)) + B*(a*b*e + b*(-(c*d) + b*e)*x - 2*a*c*(d + e*x)))))/(15*c*(b^2 - 4*a*c)^
3*(a + x*(b + c*x))^(5/2))

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Maple [B]  time = 0.007, size = 608, normalized size = 2.7 \begin{align*} -{\frac{256\,Ab{c}^{4}e{x}^{5}-512\,A{c}^{5}d{x}^{5}-128\,Ba{c}^{4}e{x}^{5}-96\,B{b}^{2}{c}^{3}e{x}^{5}+256\,Bb{c}^{4}d{x}^{5}+640\,A{b}^{2}{c}^{3}e{x}^{4}-1280\,Ab{c}^{4}d{x}^{4}-320\,Bab{c}^{3}e{x}^{4}-240\,B{b}^{3}{c}^{2}e{x}^{4}+640\,B{b}^{2}{c}^{3}d{x}^{4}+640\,Aab{c}^{3}e{x}^{3}-1280\,Aa{c}^{4}d{x}^{3}+480\,A{b}^{3}{c}^{2}e{x}^{3}-960\,A{b}^{2}{c}^{3}d{x}^{3}-320\,B{a}^{2}{c}^{3}e{x}^{3}-480\,Ba{b}^{2}{c}^{2}e{x}^{3}+640\,Bab{c}^{3}d{x}^{3}-180\,B{b}^{4}ce{x}^{3}+480\,B{b}^{3}{c}^{2}d{x}^{3}+960\,Aa{b}^{2}{c}^{2}e{x}^{2}-1920\,Aab{c}^{3}d{x}^{2}+80\,A{b}^{4}ce{x}^{2}-160\,A{b}^{3}{c}^{2}d{x}^{2}-480\,B{a}^{2}b{c}^{2}e{x}^{2}-400\,Ba{b}^{3}ce{x}^{2}+960\,Ba{b}^{2}{c}^{2}d{x}^{2}-30\,B{b}^{5}e{x}^{2}+80\,B{b}^{4}cd{x}^{2}+480\,A{a}^{2}b{c}^{2}ex-960\,A{a}^{2}{c}^{3}dx+240\,Aa{b}^{3}cex-480\,Aa{b}^{2}{c}^{2}dx-10\,A{b}^{5}ex+20\,A{b}^{4}cdx-480\,B{a}^{2}{b}^{2}cex+480\,B{a}^{2}b{c}^{2}dx-40\,Ba{b}^{4}ex+240\,Ba{b}^{3}cdx-10\,B{b}^{5}dx+192\,A{a}^{3}{c}^{2}e+96\,A{a}^{2}{b}^{2}ce-480\,A{a}^{2}b{c}^{2}d-4\,Aa{b}^{4}e+80\,Aa{b}^{3}cd-6\,A{b}^{5}d-192\,B{a}^{3}bce+192\,B{a}^{3}{c}^{2}d-16\,B{a}^{2}{b}^{3}e+96\,B{a}^{2}{b}^{2}cd-4\,Ba{b}^{4}d}{960\,{a}^{3}{c}^{3}-720\,{a}^{2}{b}^{2}{c}^{2}+180\,a{b}^{4}c-15\,{b}^{6}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x)

[Out]

-2/15/(c*x^2+b*x+a)^(5/2)*(128*A*b*c^4*e*x^5-256*A*c^5*d*x^5-64*B*a*c^4*e*x^5-48*B*b^2*c^3*e*x^5+128*B*b*c^4*d
*x^5+320*A*b^2*c^3*e*x^4-640*A*b*c^4*d*x^4-160*B*a*b*c^3*e*x^4-120*B*b^3*c^2*e*x^4+320*B*b^2*c^3*d*x^4+320*A*a
*b*c^3*e*x^3-640*A*a*c^4*d*x^3+240*A*b^3*c^2*e*x^3-480*A*b^2*c^3*d*x^3-160*B*a^2*c^3*e*x^3-240*B*a*b^2*c^2*e*x
^3+320*B*a*b*c^3*d*x^3-90*B*b^4*c*e*x^3+240*B*b^3*c^2*d*x^3+480*A*a*b^2*c^2*e*x^2-960*A*a*b*c^3*d*x^2+40*A*b^4
*c*e*x^2-80*A*b^3*c^2*d*x^2-240*B*a^2*b*c^2*e*x^2-200*B*a*b^3*c*e*x^2+480*B*a*b^2*c^2*d*x^2-15*B*b^5*e*x^2+40*
B*b^4*c*d*x^2+240*A*a^2*b*c^2*e*x-480*A*a^2*c^3*d*x+120*A*a*b^3*c*e*x-240*A*a*b^2*c^2*d*x-5*A*b^5*e*x+10*A*b^4
*c*d*x-240*B*a^2*b^2*c*e*x+240*B*a^2*b*c^2*d*x-20*B*a*b^4*e*x+120*B*a*b^3*c*d*x-5*B*b^5*d*x+96*A*a^3*c^2*e+48*
A*a^2*b^2*c*e-240*A*a^2*b*c^2*d-2*A*a*b^4*e+40*A*a*b^3*c*d-3*A*b^5*d-96*B*a^3*b*c*e+96*B*a^3*c^2*d-8*B*a^2*b^3
*e+48*B*a^2*b^2*c*d-2*B*a*b^4*d)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.18139, size = 1030, normalized size = 4.58 \begin{align*} \frac{{\left ({\left (2 \,{\left (4 \,{\left (\frac{2 \,{\left (8 \, B b c^{4} d - 16 \, A c^{5} d - 3 \, B b^{2} c^{3} e - 4 \, B a c^{4} e + 8 \, A b c^{4} e\right )} x}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}} + \frac{5 \,{\left (8 \, B b^{2} c^{3} d - 16 \, A b c^{4} d - 3 \, B b^{3} c^{2} e - 4 \, B a b c^{3} e + 8 \, A b^{2} c^{3} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x + \frac{5 \,{\left (24 \, B b^{3} c^{2} d + 32 \, B a b c^{3} d - 48 \, A b^{2} c^{3} d - 64 \, A a c^{4} d - 9 \, B b^{4} c e - 24 \, B a b^{2} c^{2} e + 24 \, A b^{3} c^{2} e - 16 \, B a^{2} c^{3} e + 32 \, A a b c^{3} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x + \frac{5 \,{\left (8 \, B b^{4} c d + 96 \, B a b^{2} c^{2} d - 16 \, A b^{3} c^{2} d - 192 \, A a b c^{3} d - 3 \, B b^{5} e - 40 \, B a b^{3} c e + 8 \, A b^{4} c e - 48 \, B a^{2} b c^{2} e + 96 \, A a b^{2} c^{2} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x - \frac{5 \,{\left (B b^{5} d - 24 \, B a b^{3} c d - 2 \, A b^{4} c d - 48 \, B a^{2} b c^{2} d + 48 \, A a b^{2} c^{2} d + 96 \, A a^{2} c^{3} d + 4 \, B a b^{4} e + A b^{5} e + 48 \, B a^{2} b^{2} c e - 24 \, A a b^{3} c e - 48 \, A a^{2} b c^{2} e\right )}}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}\right )} x - \frac{2 \, B a b^{4} d + 3 \, A b^{5} d - 48 \, B a^{2} b^{2} c d - 40 \, A a b^{3} c d - 96 \, B a^{3} c^{2} d + 240 \, A a^{2} b c^{2} d + 8 \, B a^{2} b^{3} e + 2 \, A a b^{4} e + 96 \, B a^{3} b c e - 48 \, A a^{2} b^{2} c e - 96 \, A a^{3} c^{2} e}{b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}}}{15 \,{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

1/15*(((2*(4*(2*(8*B*b*c^4*d - 16*A*c^5*d - 3*B*b^2*c^3*e - 4*B*a*c^4*e + 8*A*b*c^4*e)*x/(b^6*c^3 - 12*a*b^4*c
^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6) + 5*(8*B*b^2*c^3*d - 16*A*b*c^4*d - 3*B*b^3*c^2*e - 4*B*a*b*c^3*e + 8*A*b^2*
c^3*e)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 5*(24*B*b^3*c^2*d + 32*B*a*b*c^3*d - 48*A*b
^2*c^3*d - 64*A*a*c^4*d - 9*B*b^4*c*e - 24*B*a*b^2*c^2*e + 24*A*b^3*c^2*e - 16*B*a^2*c^3*e + 32*A*a*b*c^3*e)/(
b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 5*(8*B*b^4*c*d + 96*B*a*b^2*c^2*d - 16*A*b^3*c^2*d
- 192*A*a*b*c^3*d - 3*B*b^5*e - 40*B*a*b^3*c*e + 8*A*b^4*c*e - 48*B*a^2*b*c^2*e + 96*A*a*b^2*c^2*e)/(b^6*c^3 -
12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - 5*(B*b^5*d - 24*B*a*b^3*c*d - 2*A*b^4*c*d - 48*B*a^2*b*c^2*d
+ 48*A*a*b^2*c^2*d + 96*A*a^2*c^3*d + 4*B*a*b^4*e + A*b^5*e + 48*B*a^2*b^2*c*e - 24*A*a*b^3*c*e - 48*A*a^2*b*
c^2*e)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - (2*B*a*b^4*d + 3*A*b^5*d - 48*B*a^2*b^2*c*d
- 40*A*a*b^3*c*d - 96*B*a^3*c^2*d + 240*A*a^2*b*c^2*d + 8*B*a^2*b^3*e + 2*A*a*b^4*e + 96*B*a^3*b*c*e - 48*A*a
^2*b^2*c*e - 96*A*a^3*c^2*e)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))/(c*x^2 + b*x + a)^(5/2)