3.346 \(\int (d+e x)^{5/2} (b x+c x^2)^2 \, dx\)
Optimal. Leaf size=147 \[ \frac{2 (d+e x)^{11/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{11 e^5}+\frac{2 d^2 (d+e x)^{7/2} (c d-b e)^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}-\frac{4 d (d+e x)^{9/2} (c d-b e) (2 c d-b e)}{9 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5} \]
[Out]
(2*d^2*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^5) + (2*(
6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^5) +
(2*c^2*(d + e*x)^(15/2))/(15*e^5)
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Rubi [A] time = 0.0651057, antiderivative size = 147, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used =
{698} \[ \frac{2 (d+e x)^{11/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{11 e^5}+\frac{2 d^2 (d+e x)^{7/2} (c d-b e)^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}-\frac{4 d (d+e x)^{9/2} (c d-b e) (2 c d-b e)}{9 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]
[Out]
(2*d^2*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^5) + (2*(
6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^5) +
(2*c^2*(d + e*x)^(15/2))/(15*e^5)
Rule 698
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin{align*} \int (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac{d^2 (c d-b e)^2 (d+e x)^{5/2}}{e^4}+\frac{2 d (c d-b e) (-2 c d+b e) (d+e x)^{7/2}}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{11/2}}{e^4}+\frac{c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac{2 d^2 (c d-b e)^2 (d+e x)^{7/2}}{7 e^5}-\frac{4 d (c d-b e) (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}\\ \end{align*}
Mathematica [A] time = 0.0776933, size = 124, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (65 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 b c e \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )+c^2 \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]
[Out]
(2*(d + e*x)^(7/2)*(65*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*b*c*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^
2 + 231*e^3*x^3) + c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5
)
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Maple [A] time = 0.049, size = 141, normalized size = 1. \begin{align*}{\frac{6006\,{c}^{2}{x}^{4}{e}^{4}+13860\,bc{e}^{4}{x}^{3}-3696\,{c}^{2}d{e}^{3}{x}^{3}+8190\,{b}^{2}{e}^{4}{x}^{2}-7560\,bcd{e}^{3}{x}^{2}+2016\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-3640\,{b}^{2}d{e}^{3}x+3360\,bc{d}^{2}{e}^{2}x-896\,{c}^{2}{d}^{3}ex+1040\,{b}^{2}{d}^{2}{e}^{2}-960\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(5/2)*(c*x^2+b*x)^2,x)
[Out]
2/45045*(e*x+d)^(7/2)*(3003*c^2*e^4*x^4+6930*b*c*e^4*x^3-1848*c^2*d*e^3*x^3+4095*b^2*e^4*x^2-3780*b*c*d*e^3*x^
2+1008*c^2*d^2*e^2*x^2-1820*b^2*d*e^3*x+1680*b*c*d^2*e^2*x-448*c^2*d^3*e*x+520*b^2*d^2*e^2-480*b*c*d^3*e+128*c
^2*d^4)/e^5
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Maxima [A] time = 1.13589, size = 188, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{2} - 6930 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 4095 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 10010 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
2/45045*(3003*(e*x + d)^(15/2)*c^2 - 6930*(2*c^2*d - b*c*e)*(e*x + d)^(13/2) + 4095*(6*c^2*d^2 - 6*b*c*d*e + b
^2*e^2)*(e*x + d)^(11/2) - 10010*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d)^(9/2) + 6435*(c^2*d^4 - 2*b*c
*d^3*e + b^2*d^2*e^2)*(e*x + d)^(7/2))/e^5
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Fricas [A] time = 1.90238, size = 564, normalized size = 3.84 \begin{align*} \frac{2 \,{\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e + 520 \, b^{2} d^{5} e^{2} + 231 \,{\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \,{\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \, b^{2} e^{7}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 299 \, b^{2} d e^{6}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 1469 \, b^{2} d^{2} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 65 \, b^{2} d^{3} e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{6} e - 60 \, b c d^{5} e^{2} + 65 \, b^{2} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
2/45045*(3003*c^2*e^7*x^7 + 128*c^2*d^7 - 480*b*c*d^6*e + 520*b^2*d^5*e^2 + 231*(31*c^2*d*e^6 + 30*b*c*e^7)*x^
6 + 63*(71*c^2*d^2*e^5 + 270*b*c*d*e^6 + 65*b^2*e^7)*x^5 + 35*(c^2*d^3*e^4 + 318*b*c*d^2*e^5 + 299*b^2*d*e^6)*
x^4 - 5*(8*c^2*d^4*e^3 - 30*b*c*d^3*e^4 - 1469*b^2*d^2*e^5)*x^3 + 3*(16*c^2*d^5*e^2 - 60*b*c*d^4*e^3 + 65*b^2*
d^3*e^4)*x^2 - 4*(16*c^2*d^6*e - 60*b*c*d^5*e^2 + 65*b^2*d^4*e^3)*x)*sqrt(e*x + d)/e^5
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Sympy [B] time = 30.1276, size = 695, normalized size = 4.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(5/2)*(c*x**2+b*x)**2,x)
[Out]
2*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 4*b**2*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**3 + 2*b**2*(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**3 + 4*b*c*d**2*(-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 + 8*b*c*d*(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 + 4*b*c*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d +
e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**
(13/2)/13)/e**4 + 2*c**2*d**2*(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**5 + 4*c**2*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**
(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)
/13)/e**5 + 2*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3
*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5
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Giac [B] time = 1.35852, size = 844, normalized size = 5.74 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
2/45045*(429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*d^2*e^(-2) + 286*(35*(x*
e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b*c*d^2*e^(-3) + 13*
(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x
*e + d)^(3/2)*d^4)*c^2*d^2*e^(-4) + 286*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2
- 105*(x*e + d)^(3/2)*d^3)*b^2*d*e^(-2) + 52*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(
7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b*c*d*e^(-3) + 10*(693*(x*e + d)^(13/2) - 4095
*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*
(x*e + d)^(3/2)*d^5)*c^2*d*e^(-4) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d
^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^2*e^(-2) + 10*(693*(x*e + d)^(13/2) - 4095*(x*e +
d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d
)^(3/2)*d^5)*b*c*e^(-3) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100
100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c
^2*e^(-4))*e^(-1)